HIGH-DIMENSIONAL ELLIPSOIDS CONVERGE TO GAUSSIAN … · CONVERGE TO GAUSSIAN SPACES DAISUKE KAZUKAWA AND TAKASHI SHIOYA Abstract. We prove the convergence of (solid) ellipsoids to

HIGH-DIMENSIONAL ELLIPSOIDS

CONVERGE TO GAUSSIAN SPACES

DAISUKE KAZUKAWA AND TAKASHI SHIOYA

Abstract We prove the convergence of (solid) ellipsoids to aGaussian space in Gromovrsquos concentrationweak topology as thedimension diverges to infinity This gives the first discovered ex-ample of an irreducible nontrivial convergent sequence in the con-centration topology where lsquoirreducible nontrivialrsquo roughly meansto be not constructed from Levy families nor box convergent se-quences

1 Introduction

The study of convergence of metric measure spaces is one of cen-tral topics in geometric analysis on metric measure spaces We referto [8 9 14 23 24] for some celebrated works on it Such the studyoriginates that of Gromov-Hausdorff convergencecollapsing of Rie-mannian manifolds which has widely been developed and applied tosolutions to many significant problems in geometry and topology in-cluding Thurstonrsquos geometrization conjecture [21 28] As the startingpoint of geometric analytic study in the collapsing theory Fukaya [5]introduced the concept of measured Gromov-Hausdorff convergence ofmetric measure spaces to study the Laplacian of collapsing Riemann-ian manifolds There he discovered that only the metric structure butalso the measure structure plays an important role in the collapsingphenomena After that Cheeger-Colding [2ndash4] established a theoryof measured Gromov-Hausdorff limits of complete Riemannian mani-folds with a lower bound of Ricci curvature which is nowadays widelyapplied in the Riemannian and Kahler geometry

Meanwhile Gromov [9 Chapter 312+

] (see also [25]) has developed anew convergence theory of metric measure spaces based on the concen-tration of measure phenomenon due to Levy and V Milman [121315]where the concentration of measure phenomenon is roughly stated asthat any 1-Lipschitz function on high-dimensional spaces is almost con-stant In Gromovrsquos theory he introduced two fundamental concepts of

Date March 12 20202010 Mathematics Subject Classification 53C23Key words and phrases ellipsoid concentration topology Gaussian space ob-

servable distance box distance pyramidThis work was supported by JSPS KAKENHI Grant Number 19K03459 and

17J021211

2 DAISUKE KAZUKAWA AND TAKASHI SHIOYA

distance functions the observable distance function dconc and the box

distance function on the set say X of isomorphism classes of metricmeasure spaces The box distance function is nearly a metrization ofmeasured Gromov-Hausdorff convergence (precisely the isomorphismclasses are little different) while the observable distance function in-duces a very characteristic topology called the concentration topologywhich is effective to capture the high-dimensional aspects of spacesThe concentration topology is weaker than the box topology and inparticular a measured Gromov-Hausdorff convergence becomes a con-vergence in the concentration topology He also introduced a naturalcompactification say Π of X with respect to the concentration topol-ogy where the topology on Π is called the weak topology The concen-tration topology is sometimes useful to investigate the dimension-freeproperties of manifolds For example it has been applied to obtain anew dimension-free estimate of eigenvalue ratios of the drifted Lapla-cian on a closed Riemannian manifold with nonnegative Bakry-EmeryRicci curvature [6]

The study of the concentration and weak topologies has been growingrapidly in recent years (see [6101116ndash202225ndash27]) However thereare only a few nontrivial examples of convergent sequences of met-ric measure spaces in the concentration and weak topologies wherelsquonontrivialrsquo means neither to be a Levy family (ie convergent to aone-point space) to infinitely dissipate (see Subsection 26 for dissi-pation) nor to be box convergent One way to construct a nontriv-ial convergent sequence is to take the disjoint union or the product(more generally the fibration) of trivial sequences and to perform lit-tle surgery on it (and also to repeat these procedures finitely manytimes) We call a sequence obtained in this way a reducible sequenceAn irreducible sequence is a sequence that is not reducible In thispaper any sequence of (solid) ellipsoids has a subsequence convergingto an infinite-dimensional Gaussian space in the concentrationweaktopology This provides a new family of nontrivial weak convergentsequences and especially contains the first discovered example of anirreducible nontrivial sequence that is convergent in the concentrationtopology

Let us state our main results precisely A solid ellipsoid and anellipsoid are respectively written as

Enαi = x isin Rn |nsum

x2iα2i

Snminus1αi = x isin Rn |

x2iα2i

HIGH-DIMENSIONAL ELLIPSOIDS 3

where αi i = 1 2 n is a finite sequence of positive real num-bers See Section 3 for the definition of their metric-measure struc-tures Denote by En

αi any one of Enαi and Snminus1αi Let us given a

sequence En(j)αijij of (solid) ellipsoids where αij i = 1 2 n(j)

j = 1 2 is a double sequence of positive real numbers Our prob-lem is to determine under what condition it will converge in the con-centrationweak topology and to describe its limit

In the case where the dimension n(j) is bounded for all j the problemis easy to solve In fact such the sequence has a Hausdorff-convergentsubsequence in a Euclidean space which is also box convergent if αijis bounded for all i and j the sequence has an infinitely dissipatingsubsequence if αij is unbounded

We set aij = αijradicn(j)minus 1 If n(j) and supi aij both diverge to

infinity as j rarr infin then it is also easy to prove that Enαi infinitely

dissipates (see Proposition 33)By the reason we have mentioned above we assume

(A0) n(j) diverges to infinity as j rarr infin and aij is bounded for all iand j

We further consider the following three conditions

(A1) n(j) is monotone nondecreasing in j(A2) aij is monotone nonincreasing in i for each j(A3) aij converges to a real number say ai as j rarr infin for each i

Note that (A2) and (A3) together imply that ai is monotone nonin-creasing in i

Any sequence of (solid) ellipsoids with (A0) contains a subsequence

Ej such that each Ej is isomorphic to En(j)

radicn(j)minus1 aiji

for some se-

quence aij satisfying (A0)ndash(A3) In fact we have a subsequencefor which the dimensions satisfy (A1) Then exchanging the axes ofcoordinate provides (A2) A diagonal argument proves to have a sub-sequence satisfying (A3) Thus our problem becomes to investigate

the convergence of En(j)

j satisfying (A0)ndash(A3)

One of our main theorems is stated as follows Refer to Subsection28 for the definition of the Gaussian space Γinfin

Theorem 11 Let aij i = 1 2 n(j) j = 1 2 be a sequence

of positive real numbers satisfying (A0)ndash(A3) Then En(j)

converges weakly to the infinite-dimensional Gaussian space Γinfina2

j rarr infin This convergence becomes a convergence in the concentration

topology if and only if ai is an l2-sequence Moreover this conver-

gence becomes an asymptotic concentration (ie a dconc-Cauchy se-

quence) if and only if ai converges to zero

Gromov presents an exercise [9 31257] which is some easier special

cases of our theorem Our theorem provides not only an answer butalso a complete generalization of his exercise

For the case of round spheres (ie aij = a1j for all i and j) and alsoof projective spaces the theorem is formerly obtained by the secondnamed author [25 26] for which the convergence is only weak Alsothe weak convergence of Stiefel and flag manifolds are studied jointlyby Takatsu and the second named author [27]

We emphasize that convergence in the weakconcentration topol-ogy is completely different from weak convergence of measures Forinstance the Prokhorov distance between the normalized volume mea-sure on Snminus1(

radicnminus 1) and the n-dimensional standard Gaussian mea-

sure on Rn is bounded away from zero [27] though they both convergeto the infinite-dimensional standard Gaussian space in Gromovrsquos weaktopology

As for the characterization of weak convergence of measures weprove in Proposition 42 that if aiji l2-converges to an l2-sequence

ai as j rarr infin then the measure of En(j)

converges weakly to

the Gaussian measure γinfinai on a Hilbert space and consequently theweak convergence in Theorem 11 becomes the box convergence Con-versely the l2-convergence of aiji is also a necessary condition forthe box convergence of the (solid) ellipsoids as is seen in the followingtheorem

of positive real numbers satisfying (A0)ndash(A3) Then the convergence

in Theorem 11 becomes a box convergence if and only if we have

infinsum

a2i lt +infin and limjrarrinfin

n(j)sum

(aij minus ai)2 = 0

Theorems 11 and 12 together provide an example of irreduciblenontrivial convergent sequence of metric measure spaces in the con-centration topology ie the sequence of the (solid) ellipsoids with anl2-sequence ai and with a non-l2-convergent aiji as j rarr infin

The proof of the lsquoonly ifrsquo part of Theorem 12 is highly nontrivial Ifaiji does not l2-converge then it is easy to see that the measure ofthe (solid) ellipsoid in such the sequence does not converge weakly inthe Hilbert space However this is not enough to obtain the box non-convergence because we consider the isomorphism classes of (solid)ellipsoids for the box convergence For the complete proof we need adelicate discussion using Theorem 11

Let us briefly mention the outline of the proof of the weak con-vergence of solid ellipsoids in Theorem 11 For simplicity we set

En = En(j)

and Γ = Γinfina2i

For the weak convergence it

is sufficient to show that

(11) the limit of En dominates Γ(12) Γ dominates the limit of En

where for two metric measure spaces X and Y the space X dominates

Y if there is a 1-Lipschitz map from X to Y preserving their measures(11) easily follows from the Maxwell-Boltzmann distribution law

(Proposition 32)(12) is much harder to prove Let us first consider the simple case

where En is the ball Bn(radicnminus 1) of radius

radicnminus 1 and where Γ = Γinfin

We see that for any fixed 0 lt θ lt 1 the n-dimensional Gaussian mea-sure γn12 and the normalized volume measure of Bn(θ

radicnminus 1) both are

very small for large n Ignoring this small part Bn(θradicnminus 1) we find

a measure-preserving isotropic map say ϕ from Γn12 Bn(θradicnminus 1)

to the annulus Bn(radicnminus 1) Bn(θ

radicnminus 1) where we normalize their

measures to be probability Estimating the Lipschitz constant of ϕwe obtain (12) with error This error is estimated and we eventuallyobtain the required weak convergence

We next try to apply this discussion for solid ellipsoids We considerthe distortion of the above isotropic map ϕ by a linear transforma-tion determined by aij However the Lipschitz constant of such thedistorted isotropic map is arbitrarily large depending on aij To over-come this problem we settle the assumptions (A0)ndash(A3) from whichthe discussion boils down to the special case where ai = aN for alli ge N and aij = ai ge aN for all i j and for a (large) number N Infact by (A0)ndash(A3) the solid ellipsoid En for large n and the Gaussianspace Γ are both close to those in the above special case In this specialcase the Gaussian measure γna2i

and the normalized volume measure

of En of the domain

x isin Rn o | |xi|x lt ε for any i = 1 N minus 1

are both almost full for large n and for any fixed ε gt 0 On this domainwe are able to estimate the Lipschitz constant of the distorted isotropicmap With some careful error estimates letting ǫ rarr 0+ and θ rarr 1minuswe prove the weak convergence of En to Γ

2 Preliminaries

In this section we survey the definitions and the facts needed in thispaper We refer to [9 Chapter 31

2+] and [25] for more details

21 Distance between measures

Definition 21 (Total variation distance) The total variation distance

dTV(micro ν) of two Borel probability measures micro and ν on a topologicalspace X is defined by

dTV(micro ν) = supA

|micro(A)minus ν(A) |

where A runs over all Borel subsets of X

If micro and ν are both absolutely continuous with respect to a Borelmeasure ω on X then

dTV(micro ν) =1

∣∣∣∣dmicro

dωminus dν

∣∣∣∣ dω

where dmicrodω

is the Radon-Nikodym derivative of micro with respect to ω

Definition 22 (Prokhorov distance) The Prokhorov distance dP(micro ν)between two Borel probability measures micro and ν on a metric space(X dX) is defined to be the infimum of ε ge 0 satisfying

micro(Bε(A)) ge ν(A)minus ε

for any Borel subset A sub X where Bε(A) = x isin X | dX(xA) lt ε The Prokhorov metric is a metrization of weak convergence of Borel

probability measures on X provided thatX is a separable metric spaceIt is known that dP le dTV

Definition 23 (Ky Fan distance) Let (X micro) be a measure space andY a metric space For two micro-measurable maps f g X rarr Y we definethe Ky Fan distance dKF(f g) between f and g to be the infimum ofε ge 0 satisfying

micro( x isin X | dY (f(x) g(x)) gt ε ) le ε

dKF is a pseudo-metric on the set of micro-measurable maps from X toY It holds that dKF(f g) = 0 if and only if f = g micro-ae We havedP(flowastmicro glowastmicro) le dKF(f g) where flowastmicro is the push-forward of micro by f

Let p be a real number with p ge 1 and (X dX) a complete separablemetric space

Definition 24 The p-Wasserstein distance between two Borel prob-ability measures micro and ν on X is defined to be

Wp(micro ν) = infπisinΠ(microν)

XtimesXdX(x x

prime)p dπ(x xprime)

(le +infin)

where Π(micro ν) is the set of couplings between micro and ν ie the set ofBorel probability measures π on X times X such that π(A times X) = micro(A)and π(X times A) = ν(A) for any Borel subset A sub X

Lemma 25 Let micro and micron n = 1 2 be Borel probability measures

on X Then the following (1) and (2) are equivalent to each other

(1) Wp(micron micro) rarr 0 as nrarr infin

(2) micron converges weakly to micro as nrarr infin and the p-th moment of micronis uniformly bounded

lim supnrarrinfin

dX(x0 x)p dmicron(x) lt +infin

for some point x0 isin X

It is known that dP2 leW1 and that Wp leWq for any 1 le p le q

22 mm-Isomorphism and Lipschitz order

Definition 26 (mm-Space) Let (X dX) be a complete separable met-ric space and microX a Borel probability measure on X We call the triple(X dX microX) an mm-space We sometimes say that X is an mm-spacein which case the metric and the Borel measure of X are respectivelyindicated by dX and microX

Definition 27 (mm-Isomorphism) Two mm-spaces X and Y are saidto be mm-isomorphic to each other if there exists an isometry f supp microX rarr supp microY with flowastmicroX = microY where supp microX is the support ofmicroX Such an isometry f is called an mm-isomorphism Denote by Xthe set of mm-isomorphism classes of mm-spaces

Note that X is mm-isomorphic to (supp microX dX microX)We assume that an mm-space X satisfies

X = supp microX

unless otherwise stated

Definition 28 (Lipschitz order) Let X and Y be two mm-spaces Wesay that X (Lipschitz ) dominates Y and write Y ≺ X if there exists a1-Lipschitz map f X rarr Y satisfying flowastmicroX = microY We call the relation≺ on X the Lipschitz order

The Lipschitz order ≺ is a partial order relation on X

23 Observable diameter The observable diameter is one of themost fundamental invariants of an mm-space up to mm-isomorphism

Definition 29 (Partial and observable diameter) Let X be an mm-space and let κ gt 0 We define the κ-partial diameter diam(X 1minusκ) =diam(microX 1 minus κ) of X to be the infimum of the diameter of A whereA sub X runs over all Borel subsets with microX(A) ge 1 minus κ Denote byLip1(X) the set of 1-Lipschitz continuous real-valued functions on X We define the (κ-)observable diameter of X by

ObsDiam(X minusκ) = supfisinLip1(X)

diam(flowastmicroX 1minus κ)

ObsDiam(X) = infκgt0

maxObsDiam(X minusκ) κ

It is easy to see that the (κ-)observable diameter is monotone non-decreasing with respect to the Lipschitz order relation

24 Box distance and observable distance

Definition 210 (Parameter) Let I = [ 0 1 ) and let X be an mm-space A map ϕ I rarr X is called a parameter of X if ϕ is a Borel mea-surable map with ϕlowastL1 = microX where L1 denotes the one-dimensionalLebesgue measure on I

It is known that any mm-space has a parameter

Definition 211 (Box distance) We define the box distance (X Y )between two mm-spaces X and Y to be the infimum of ε ge 0 satisfyingthat there exist parameters ϕ I rarr X ψ I rarr Y and a Borel subsetI sub I such that

L1(I) ge 1minus ε and |ϕlowastdX(s t)minus ψlowastdY (s t) | le ε

for any s t isin I where ϕlowastdX(s t) = dX(ϕ(s) ϕ(t)) for s t isin I

The box metric is a complete separable metric on X

Definition 212 (ε-mm-isomorphism) Let ε be a nonnegative realnumber A map f X rarr Y between two mm-spaces X and Y is calledan ε-mm-isomorphism if there exists a Borel subset X sub X such that

(i) microX(X) ge 1minus ε(ii) | dX(x xprime)minus dY (f(x) f(x

prime)) | le ε for any x xprime isin X (iii) dP(flowastmicroX microY ) le ε

We call the set X a nonexceptional domain of f

Lemma 213 Let X and Y be two mm-spaces and let ε ge 0(1) If there exists an ε-mm-isomorphism from X to Y then (X Y ) le

3ε(2) If (X Y ) le ε then there exists a 3ε-mm-isomorphism from

X to Y

Definition 214 (Observable distance) For any parameter ϕ of X weset

ϕlowastLip1(X) = f ϕ | f isin Lip1(X) We define the observable distance dconc(X Y ) between two mm-spaces

X and Y by

dconc(X Y ) = infϕψ

dH(ϕlowastLip1(X) ψlowastLip1(Y ))

where ϕ I rarr X and ψ I rarr Y run over all parameters of X and Y respectively and where dH is the Hausdorff metric with respect to theKy Fan metric for the one-dimensional Lebesgue measure on I dconcis a metric on X

It is known that dconc le and that the concentration topology isweaker than the box topology

25 Pyramid

Definition 215 (Pyramid) A subset P sub X is called a pyramid if itsatisfies the following (i)ndash(iii)

(i) If X isin P and if Y ≺ X then Y isin P(ii) For any two mm-spaces XX prime isin P there exists an mm-space

Y isin P such that X ≺ Y and X prime ≺ Y (iii) P is nonempty and box closed

We denote the set of pyramids by Π Note that Gromovrsquos definition ofa pyramid is only by (i) and (ii) (iii) is added in [25] for the Hausdorffproperty of Π

For an mm-space X we define

PX = X prime isin X | X prime ≺ X which is a pyramid We call PX the pyramid associated with X

We observe that X ≺ Y if and only if PX sub PY It is trivial thatX is a pyramid

We have a metric denoted by ρ on Π for which we omit to statethe definition We say that a sequence of pyramids converges weakly

to a pyramid if it converges with respect to ρ We have the following

(1) The map ι X ni X 7rarr PX isin Π is a 1-Lipschitz topologicalembedding map with respect to dconc and ρ

(2) Π is ρ-compact(3) ι(X ) is ρ-dense in Π

In particular (Π ρ) is a compactification of (X dconc) We say that asequence of mm-spaces converges weakly to a pyramid if the associatedpyramid converges weakly Note that we identify X with PX in Section1

For an mm-space X a pyramid P and t gt 0 we define

tX = (X t dX microX) and tP = tX | X isin P We see P tX = tPX It is easy to see that tP is continuous in t withrespect to ρ

We have the following

Proposition 216 For any two Borel probability measures micro and νon a complete separable metric space X we have

ρ(P(X micro)P(X ν)) le dconc((X micro) (X ν)) le ((X micro) (X ν))

le 2 dP(micro ν) le 2 dTV(micro ν)

26 Dissipation Dissipation is the opposite notion to concentrationWe omit to state the definition of the infinite dissipation Instead westate the following proposition Let Xn n = 1 2 be a sequenceof mm-spaces

Proposition 217 The sequence Xn infinitely dissipates if and only

if PXn converges weakly to X as nrarr infin

An easy discussion using [19 Lemma 66] leads to the following

Proposition 218 The following (1) and (2) are equivalent to each

(1) The κ-observable diameter ObsDiam(Xnminusκ) diverges to infin-

ity as nrarr infin for any κ isin ( 0 1 )(2) Xn infinitely dissipates

27 Asymptotic concentration We say that a sequence of mm-spaces asymptotically concentrates if it is a dconc-Cauchy sequence Itis known that any asymptotically concentrating sequence convergesweakly to a pyramid A pyramid P is said to be concentrated if(Lip1(X) sim dKF)XisinP is precompact with respect to the Gromov-Hausdorff distance where f sim g holds if f minus g is constant

Theorem 219 Let P be a pyramid The following (1)ndash(3) are equiv-

alent to each other

(1) P is concentrated

(2) There exists a sequence of mm-spaces asymptotically concen-

trating to P

(3) If a sequence of mm-spaces converges weakly to P then it asymp-

totically concentrates

28 Gaussian space Let ai i = 1 2 n be a finite sequence ofnonnegative real numbers The product

γna2i =

notimes

γ1a2i

of the one-dimensional centered Gaussian measure γ1a2i

of variance a2iis an n-dimensional centered Gaussian measure on Rn where we agreethat γ102 is the Dirac measure at 0 and γna2i

is possibly degenerate We

call the mm-space Γna2i = (Rn middot γna2

i) the n-dimensional Gaussian

space with variance a2i Note that for any Gaussian measure γ onRn the mm-space (Rn middot γ) is mm-isomorphic to Γna2i

where a2i are

the eigenvalues of the covariance matrix of γWe now take an infinite sequence ai i = 1 2 of nonnegative

real numbers For 1 le k le n we denote by πnk Rn rarr Rk the naturalprojection ie

πnk (x1 x2 xn) = (x1 x2 xk) (x1 x2 xn) isin Rn

Since the projection πnnminus1 Γna2i rarr Γnminus1

a2i is 1-Lipschitz continuous

and measure-preserving for any n ge 2 the Gaussian space Γna2i is

monotone nondecreasing in n with respect to the Lipschitz order so

that as n rarr infin the associated pyramid PΓna2i converges weakly to

the -closure of⋃infinn=1PΓna2i

denoted by PΓinfina2i

We call PΓinfina2i

virtual Gaussian space with variance a2i We remark that the infiniteproduct measure

γinfina2i =

infinotimes

γ1a2i

is a Borel probability measure on Rinfin with respect to the product topol-ogy but is not necessarily Borel with respect to the l2-norm Only inthe case where

(21)infinsum

a2i lt +infin

the measure γinfina2i is a Borel measure with respect to the l2-norm middot

which is supported in the separable Hilbert space H = x isin Rinfin |x lt +infin (cf [1 sect23]) and consequently Γinfin

a2i = (H middot γinfina2i ) is

an mm-space In the case of (21) the variance of γinfina2i satisfies

x2 dγinfina2i (x) =infinsum

3 Weak convergence of ellipsoids

In this section we prove Theorem 11 We also prove the convergenceof Gaussian spaces as a corollary to the theorem

Let αi i = 1 2 n be a sequence of positive real numbers Then-dimensional solid ellipsoid En and the (n minus 1)-dimensional ellipsoidSnminus1 (defined in Section 1) are respectively obtained as the image ofthe closed unit ball Bn(1) and the unit sphere Snminus1(1) in Rn by thelinear isomorphism Lnαi R

n rarr Rn defined by

Lnαi(x) = (α1x1 αnxn) x = (x1 xn) isin Rn

We assume that the n-dimensional solid ellipsoid Enαi is equipped withthe restriction of the Euclidean distance function and with the nor-

malized Lebesgue measure ǫnαi = Ln|Enαi

where micro = micro(X)minus1micro is

the normalization of a finite measure micro on a space X and Ln the n-dimensional Lebesgue measure on Rn The (nminus1)-dimensional ellipsoidSnminus1αi is assumed to be equipped with the restriction of the Euclidean

distance function and with the push-forward σnminus1αi = (Lnαi)lowastσ

nminus1 of

the normalized volume measure σnminus1 on the unit sphere Snminus1(1) in RnThroughout this paper let (En

αi enαi) be any one of

(Enαi ǫnαi) and (Snminus1

αi σnminus1αi)

for any n ge 2 and αi The measure enαi is sometimes considered asa Borel measure on Rn supported on En

Lemma 31 Let αi and βi i = 1 2 n be two sequences of

positive real numbers If αi le βi for all i = 1 2 n then Enαi is

dominated by Enβi

Proof The map Lnαiβi Enβi rarr En

αi is 1-Lipschitz continuous andpreserves their measures

Proposition 32 (Maxwell-Boltzmann distribution law) Let aiji = 1 2 n(j) j = 1 2 be a sequence of positive real numbers

satisfying (A0) and (A3) Then (πn(j)k )lowaste

converges weakly

to γkai as j rarr infin for any fixed positive integer k where πnk is defined

in Subsection 28

Proof The proposition follows from a straightforward and standardcalculation (see [25 Proposition 21])

Proposition 33 Let aij i = 1 2 n(j) j = 1 2 be a

sequence of positive real numbers If supi aij diverges to infinity as

j rarr infin then En(j)

infinitely dissipates

Proof Assume that supi aij diverges to infinity as j rarr infin Exchangingthe coordinates we assume that a1j diverges to infinity as j rarr infinWe take any positive real number a and fix it Let aij = minaij aNote that a1j = a for all sufficiently large j By Lemma 31 the 1-

Lipschitz continuity of πn(j)1 and the Maxwell-Boltzmann distribution

law (Proposition 32) we have

lim infjrarrinfin

ObsDiam(En(j)

minusκ)

ge lim infjrarrinfin

ObsDiam(En(j)

minusκ)

ge limjrarrinfin

diam((πn(j)1 )lowaste

1minus κ)

= diam(γ1a 1minus κ)

which diverges to infinity as a rarr infin Proposition 218 leads us to the

dissipation property for En(j)

Let ai i = 1 2 n be a sequence of positive real numbers Letus construct a transport map from γna2i

to ǫnradicnminus1 ai For r ge 0 we

determine a real number R = R(r) in such a way that 0 le R leradicnminus 1

and γn12(Br(o)) = ǫnradicnminus1

(BR(o)) where ǫnradicnminus1

denotes the normalized

Lebesgue measure on Enradicnminus1

= Bradicnminus1(o) sub Rn Define an isotropic

map ϕ Rn rarr Enradicnminus1

ϕ(x) =R(x)x x x isin Rn

We remark that ϕlowastγn12 = ǫnradic

nminus1 It holds that

R = (nminus 1)12

Inminus1

tnminus1eminust2

int infin

tmeminust2

Note that R is strictly monotone increasing in r Let L = Lnai and

r = r(x) = Lminus1(x) We define

ϕE = L ϕ Lminus1 Rn rarr Enradicnminus1ai

The map ϕE is a transport map from γna2i to ǫn

radicnminus1 ai ie ϕ

Elowastγ

ǫnradicnminus1 ai It holds that ϕE(x) = R

rx if x 6= o We denote by ϕS

Rn o rarr Snminus1radicnminus1 ai the central projection with center o ie

ϕS(x) =

radicnminus 1

rx x isin Rn o

which is a transport map from γna2i to σnminus1

radicnminus1ai

For an integer N with 1 le N le n and for ε gt 0 we define

DnNε = x isin Rn o | |xj |x lt ε for any j = 1 N minus 1

For 0 lt θ lt 1 let

F nθ = x isin Rn | Lminus1(x) ge θ

radicn

Lemma 34 We assume that

(i) ai ge a for any i = 1 2 n(ii) ai = a for any i with N le i le n and for a positive integer N

with N le n

Then there exists a universal positive real number C such that for

any two real numbers θ and ε with 0 lt θ lt 1 and 0 lt ε le 1N the

operator norms of the differentials of ϕE and ϕS satisfy

dϕEx le

radic1 + CNε

θand dϕS

x leradic1 + CNε

for any x isin DnNε cap F n

Proof Let x isin DnNε capF n

θ be any point We first estimate dϕEx Take

any unit vector v isin Rn We see that

dϕEx(v)2 =

)partr

partxj〈x v〉+ R

〈x v〉2nsum

x2ja4j

r2part

)〈x v〉

vjxja2j

It follows from (i) (ii) and x isin DnNε that

x2 = 1 +Nminus1sum

a2jminus 1

x2 = 1 +O(Nε2)

and so

x = 1 +O(Nε2)xar

= 1 +O(Nε2)

We also have

x2nsum

x2ja4j

= 1 +Nminus1sum

a4jminus 1

x2 = 1 + O(Nε2)

vjxja2j

vjxjx +

Nminus1sum

a2jminus 1

)vjxjx =

〈x v〉x +O(Nε)

By these formulas setting t = 〈x v〉x and g = r partpartr

) we have

dϕEx(v)2 = t2g2(1 +O(Nε2)) +

r(1 +O(Nε2))(31)

rO(Nε) +

We are going to estimate g Letting f(r) =int r0tnminus1eminus

2 dt we have

partr=

radicnminus 1nminus1I

minus 1n

nminus1f(r)1nminus1rnminus1eminus

2 le nminus 12f(r)minus1rnminus1eminus

which together with f(r) geint r0tnminus1 dt = rn

nand r ge θ

radicn yields

0 le partR

partrle

radicn

Since R leradicnminus 1 and r ge θ

radicn we have 0 le Rr lt 1θ Therefore

|g| =∣∣∣∣partR

partrminus R

∣∣∣∣ le1

Thus (31) is reduced to

dϕEx(v)2 = t2g2 +

r2+O(θminus2Nε)

= t2(partR

+ (1minus t2)R2

r2+O(θminus2Nε)

le θminus2 +O(θminus2Nε)

This completes the required estimate of dϕEx(v)

If we replace R withradicnminus 1 then ϕE becomes ϕS and the above

formulas are all true also for ϕS This completes the proof

We now give an infinite sequence ai i = 1 2 of positive realnumbers and a positive real number a Consider the following twoconditions

(a1) ai ge a for any i(a2) ai = a for any i ge N and for a positive integer N

Lemma 35 If we assume (a2) then for any real numbers 0 lt θ lt 1and ε gt 0 we have

limnrarrinfin

enradicnminus1ai(D

nNε) = 1(1)

limnrarrinfin

γna2i (Dn

Nε cap F nθ ) = 1(2)

Proof Lemma [25 Lemma 741] tells us that γna2i (F n

θ ) = γn12(Lminus1(F n

tends to 1 as nrarr infin It holds that enradicnminus1 ai(D

nNε) = σnminus1

radicnminus1 ai

(DnNε) =

γna2i (Dn

Nε) The Maxwell-Boltzmann distribution law leads us that

σnminus1radicnminus1 ai

(DnNε) converges to 1 as n rarr infin This completes the proof

Lemma 36 Assume (a1) and (a2) If a subsequence of PEnradicnminus1 ain

converges weakly to a pyramid Pinfin as nrarr infin then

Pinfin sub PΓinfina2i

Proof Take any real number ε with 0 lt ε lt 1N and fix it Letθ = 1

radic1 + CNε where C is the constant in Lemma 34 Note that

θ satisfies 0 lt θ lt 1 and tends to 1 as εrarr 0+ We apply Lemma 34Let

ϕE if (En

radicnminus1ai e

nradicnminus1ai) = (Enradicnminus1 ai ǫ

nradicnminus1ai)

ϕS if (Enradicnminus1ai e

nradicnminus1ai) = (Snminus1

radicnminus1 ai σ

nminus1radicnminus1 ai)

Since ϕ is θminus2-Lipschitz continuous onDnNεcapF n

θ and since ϕlowast( ˜γna2i |Dn

NεcapFn

˜enradicnminus1 ai|Dn

Nε the θ2-scale change θ2Xn of the mm-space Xn = (Rn middot

˜enradicnminus1ai|Dn

Nε) is dominated by Yn = (Rn middot ˜γna2i

|DnNε

capFnθ) and so

θ2PXn = Pθ2Xn sub PYn for any n Combining Lemma 35 with Propo-sition 216 we see that as nrarr infin

ρ(θ2PXnPEnradicnminus1ai) le 2 dTV( ˜en

radicnminus1ai|Dn

Nε en

radicnminus1 ai) rarr 0

ρ(PYnPΓna2i ) le 2 dTV( ˜γna2i

|DnNε

capFnθ γna2i

) rarr 0

Therefore θ2Pinfin is contained in PΓinfina2i

As ε rarr 0+ we have θ rarr 1

and θ2Pinfin rarr Pinfin This completes the proof

Lemma 37 If we assume (a2) then PEnradicnminus1 ai converges weakly to

PΓinfina2i

as nrarr infin

Proof Assume (a2) and suppose that PEnradicnminus1 ai does not converge

weakly to PΓinfina2i

as n rarr infin Then there is a subsequence n(j)of n such that PEn(j)

radicn(j)minus1 ai

converges weakly to a pyramid Pinfin

different from PΓinfina2i

The Maxwell-Boltzmann distribution law tells us that the push-

forward measure νkn(j) = (πn(j)k )lowaste

radicn(j)minus1 ai

converges weakly to γkai

as j rarr infin for any k so that (Rk middot νkn(j)) box converges to Γka2i

Since En(j)

radicn(j)minus1 ai

dominates (Rk middot νkn(j)) the limit pyramid Pinfin

contains Γka2i for any k This proves

(32) Pinfin sup PΓinfina2i

Let ai = maxai a It follows from ai le ai that Enradicnminus1 ai is

dominated by Enradicnminus1 ai which implies PEn

radicnminus1 ai sub PEn

radicnminus1 ai for

any n By applying Lemma 36 the limit of any weakly convergentsequence of PEn

radicnminus1 ain is contained in PΓinfin

a2i Therefore Pinfin is

contained in PΓinfina2i

Denote by l the number of irsquos with ai lt a For

any k ge N we consider the projection from Γk+la2i to Γka2i

dropping

the axes xi with ai lt a which is 1-Lipschitz continuous and preservestheir measures This shows that Γk+la2

i dominates Γka2i

and so PΓinfina2i

supPΓinfin

a2i We thus obtain

(33) Pinfin sub PΓinfina2i

Combining (32) and (33) yields Pinfin = PΓinfina2i

which is a contra-

diction This completes the proof

Lemma 38 Let aij satisfy (A0)ndash(A3) If PEn(j)

converges

weakly to a pyramid Pinfin as j rarr infin then

Pinfin sup PΓinfina2i

Proof Note that the sequence ai is monotone nonincreasing Puti0 = sup i | ai gt 0 (le infin) We see ai0 gt 0 if i0 ltinfin The Maxwell-

Boltzmann distribution law proves that νkn(j) = (πn(j)k )lowaste

converges weakly to γka2i as j rarr infin for each finite k with 1 le k le i0

The ellipsoid En(j)

dominates (Rk middot νkn(j)) which converges

to Γka2i so that Γka2i

belongs to Pinfin Since Γka2i for any k ge i0 is

mm-isomorphic to Γi0a2i provided i0 ltinfin we obtain the lemma

Lemma 39 Let aij satisfy (A0)ndash(A3) If PEn(j)minus1

converges

Proof Since ai is monotone nonincreasing it converges to a nonneg-ative real number say ainfin

We first assume that ainfin gt 0 We see that ai gt 0 for any i For anyε gt 0 there is a number I(ε) such that

(34) ai le (1 + ε)ainfin for any i ge I(ε)

Also there is a number J(ε) such that

(35) aij le ai + ainfinε for any i le I(ε) and j ge J(ε)

By the monotonicity of aij in i (34) and (35) we have(36)aij le aI(ε)j le aI(ε) + ainfinε le (1 + 2ε)ainfin for any i ge I(ε) and j ge J(ε)

It follows from (35) and ainfin le ai that

(37) aij le ai + ainfinε le (1 + ε)ai for any i le I(ε) and j ge J(ε)

bεi =

ai if i le I(ε)

ainfin if i gt I(ε)

By (36) and (37) for any i and j ge J(ε) we see that aij le (1+2ε)bεiand so E

n(j)aiji ≺ E

n(j)(1+2ε)bεi = (1 + 2ε)E

n(j)bεi Lemma 37 implies that

PEn(j)

radicn(j)minus1 bεi

converges weakly to PΓinfinb2εi

as j rarr infin Therefore Pinfin

is contained in (1 + 2ε)PΓinfinb2

εi for any ε gt 0 Since bεi le ai we see

that Pinfin is contained in (1 + 2ε)PΓinfina2i

for any ε gt 0 This proves the

lemma in this caseWe next assume ainfin = 0 For any ε gt 0 there is a number I(ε) such

(38) ai lt ε for any i ge I(ε)

We may assume that I(ε) = i0 + 1 if i0 ltinfin where i0 = sup i | ai gt0 Also there is a number J(ε) such that

aI(ε)j lt aI(ε) + ε for any j ge J(ε)(39)

aij lt (1 + ε)ai for any i lt I(ε) and j ge J(ε)(310)

It follows from (38) and (39) that

(311) aij le aI(ε)j lt aI(ε) + ε lt 2ε for any i ge I(ε) and j ge J(ε)

bεi =

(1 + ε)ai if i lt I(ε)

2ε if i ge I(ε)

From (310) and (311) we have aij lt bεi for any i and j ge J(ε) and so

En(j)aiji ≺ E

n(j)bεi for j ge J(ε) Lemma 37 implies that PEn(j)

as j rarr infin Therefore Pinfin is contained

in PΓinfinb2εi

for any ε gt 0 Let k be any number with k ge I(ε) The

Gaussian space Γkb2εiis mm-isomorphic to the l2-product of Γ

I(ε)minus1

(1+ε)2a2i

and ΓkminusI(ε)+1

(2ε)2 It follows from the Gaussian isoperimetry that

ObsDiam(ΓkminusI(ε)+1(2ε)2 ) = inf

κgt0max2ε diam(γ112 1minus κ) κ = τ(ε)

which tends to zero as ε rarr 0+ If τ(ε) lt 12 then by [25 Proposition732]

ρ(PΓkb2εiPΓ

I(ε)minus1

(1+ε)2a2i ) le dconc(Γ

kb2εi

ΓI(ε)minus1

(1+ε)2a2i ) le τ(ε)

Taking the limit as k rarr infin yields

ρ(PΓinfinb2εi

PΓI(ε)minus1

(1+ε)2a2i) le τ(ε)

There is a sequence ε(l) l = 1 2 of positive real numbers tendingto zero such that PΓinfin

b2ε(l)i

converges weakly to a pyramid P primeinfin as l rarr

infin P primeinfin contains Pinfin and PΓ

I(ε(l))minus1

(1+ε(l))2a2i converges weakly to P prime

infin as

l rarr infin Since PΓI(ε(l))minus1

(1+ε(l))2a2i is contained in PΓinfin

(1+ε(l))2a2i and since

PΓinfin(1+ε(l))2a2i

= (1+ε(l))PΓinfina2i

converges weakly to PΓinfina2i

as l rarr infin

the pyramid P primeinfin is contained in PΓinfin

a2i so that Pinfin is contained in

PΓinfina2i

This completes the proof

Proof of Theorem 11 Suppose that PEn(j)

does not converge

as j rarr infin Then taking a subsequence of j we

may assume that PEn(j)

which contradicts Lemmas 38 and 39 Thus

PEn(j)

converges weakly to PΓinfina2

i as j rarr infin

As is mentioned in Subsection 28 the infinite-dimensional Gaussianspace Γinfin

a2i is well-defined as an mm-space if and only if ai is an

l2-sequence only in which case the above sequence of (solid) ellipsoidsbecomes a convergent sequence in the concentration topology

Assume that ai converges to zero as i rarr infin It is well-known thatthe Ornstein-Uhlenbeck operator (or the drifted Laplacian) on Γ1

a2 hascompact resolvent and spectrum kaminus2 | k = 0 1 2 Thus thesame proof as in [25 Corollary 735] yields that Γna2i

asymptotically

(spectrally) concentrates to Γinfinai

Conversely we assume that ai is bounded away from zero and seta = inf i ai Applying [25 Proposition 737] yields that PΓinfin

a2 is not

concentrated Since PΓinfina2

i contains PΓinfin

a2 the pyramid PΓinfina2

i is not

concentrated which implies that En(j)

does not asymptotically

concentrate (see Theorem 219)This completes the proof of the theorem

Let us next consider the convergence of the Gaussian spaces

sequence of nonnegative real numbers If supi aij diverges to infinity as

j rarr infin then Γn(j)

a2ijinfinitely dissipates

Proof Exchanging the coordinates we assume that a1j diverges to in-

finity as j rarr infin Since Γ1a21j

is dominated by Γn(j)

a2ij we have

ObsDiam(Γn(j)

a2ijminusκ) ge diam(Γ1

a21j 1minus κ) rarr infin as j rarr infin

This together with Proposition 218 completes the proof

In a similar way as in the proof of Theorem 11 we obtain the fol-lowing

Corollary 311 Let aij satisfy (A0)ndash(A3) Then Γn(j)

a2ijconverges

as j rarr infin This convergence becomes a convergence in

the concentration topology if and only if ai is an l2-sequence More-

over this convergence becomes an asymptotic concentration if and only

if ai converges to zero

Proof Suppose that Γn(j)

a2ijdoes not converge weakly to PΓinfin

a2i as j rarr

infin Then there is a subsequence of PΓn(j)

a2ijj that converges weakly

to a pyramid Pinfin different from PΓinfina2i

We write such a subsequence

by the same notation PΓn(j)

Since Γka2ijis dominated by Γ

a2ijfor k le n(j) and Γka2ij

converges

weakly to Γka2i as j rarr infin we see that Γka2i

belongs to Pinfin for any k

so that

We prove Pinfin sub PΓinfina2i

in the case of ainfin gt 0 where ainfin = limirarrinfin ai

Under ainfin gt 0 the same discussion as in the proof of Lemma 38 provesthat there are two numbers I(ε) and J(ε) for any ε gt 0 such that (36)and (37) both hold We therefore see that for any k and j ge J(ε)

Γka2ijis dominated by (1 + ε)Γka2i

and so PΓn(j)

a2ijisub (1 + ε)PΓinfin

This proves Pinfin sub PΓinfina2i

We next prove Pinfin sub PΓinfina2i

in the case of ainfin = 0 Let bεi be as

in Lemma 39 The discussion in the proof of Lemma 39 yields thataij lt bεi for any i and for every sufficiently large j which impliesPinfin sub PΓinfin

b2εi We obtain Pinfin sub PΓinfin

a2i in the same way as in the

proof of Lemma 39 The weak convergence of Γn(j)

a2ijto PΓinfin

been provedThe rest is identical to the proof of Theorem 11 This completes the

4 Box convergence of ellipsoids

The main purpose of this section is to prove Theorem 12Let us first prove the weak convergence of en

radicnminus1aij if aij l2-

converges

Lemma 41 Let A be a family of sequences of positive real numbers

such that

supaiisinA

infinsum

a2i lt +infin

Then we have

limnrarrinfin

supaiisinA

dP(ǫnradicnminus1 ai γ

) = 0(1)

lim supnrarrinfin

supaiisinA

W2(σnminus1radicnminus1 ai γ

)2 leradic2eminus1 sup

aiisinA

infinsum

a2i(2)

for any positive integer k

Proof We prove (1) Let r(x) = Lminus1(x) as in Section 3 Take anyreal number θ with 0 lt θ lt 1 and fix it Let us consider the normaliza-tion of the measures ǫn

radicnminus1 ai|rminus1([ θ

radicnminus1

radicnminus1 ]) and γ

|rminus1([ θradicnminus1θminus1

radicnminus1 ])

which we denote by ǫnθ and γnθ respectively Set

vθn = ǫnradicnminus1( x isin Rn | θradicnminus 1 le x le

radicnminus 1 )

wθn = γn12( x isin Rn | θradicnminus 1 le x le θminus1

radicnminus 1 )

We remark that

vθn = ǫnradicnminus1 ai(r

minus1([ θradicnminus 1

radicnminus 1 ]))

wθn = γna2i (rminus1([ θ

radicnminus 1 θminus1

radicnminus 1 ]))

limnrarrinfin

vθn = limnrarrinfin

wθn = 1

It then holds that

dP(ǫnradicnminus1ai ǫ

nθ ) le dTV(ǫ

nradicnminus1 ai ǫ

nθ ) = 1minus vθn(41)

dP(γna2i

γnθ ) le dTV(γna2i

γnθ ) = 1minus wθn(42)

To estimate dP(ǫnθ γ

nθ ) we define a transport map say ψ from γnθ to

ǫnθ in the same manner as for ϕE in Section 3 which is expressed as

ψ(x) =R

rx x isin rminus1([ θ

radicnminus 1 ])

where R is the function of variable r isin [ θradicnminus 1 θminus1

radicnminus 1 ] defined

θradicnminus 1 le R le

radicnminus 1 and γnθ (Br(o)) = ǫnθ (BR(o))

Since θ2 le Rr le θminus1 we have

W2(ǫnθ γ

2 leint

ψ(x)minus x 2 dγnθ (x) =int

rminus 1

x2 dγnθ (x)

le max(1minus θ)2 (θ2 minus 1)2int

x2 dγnθ (x)

le max(1minus θ)2 (θ2 minus 1)2γna2i

(rminus1([ θradicnminus 1 θminus1

radicnminus 1 ]))

x2 dγna2i (x)

=max(1minus θ)2 (θ2 minus 1)2

which together with (41) and (42) implies

dP(ǫnradicnminus1ai γ

le 2minus vθn minus wθn +

(max(1minus θ)2 (θ2 minus 1)2

and hence

lim supnrarrinfin

supaiisinA

le(max(1minus θ)2 (θ2 minus 1)2 sup

aiisinA

infinsum

rarr 0 as θ rarr 1+

This proves (1)

We prove (2) Using the transport map ϕS from γna2i to σnminus1

radicnminus1ai

in Section 3 we have

)2 leint

∥∥ z minus ϕS(z)∥∥2 dγna2i (z)

Snminus1(1)

a2ix2i dσ

nminus1(x) middot 1

Inminus1

int infin

(r minusradicnminus 1)2rnminus1eminusr

where Im =intinfin0tmeminust

22 dt We see in the proof of [25 Lemma 741]

that rmeminusr22 le mm2eminusm2eminus(rminusradic

m)22 and also that Im sim radicπ(m minus

1)m2eminus(mminus1)2 Therefore

Inminus1

int infin

22 dr le 1

Inminus1

radic2π(nminus 1)(nminus1)2eminus(nminus1)2

simradic2eminus12

(1minus 1(nminus 1))(nminus1)2minusrarr

radic2eminus1 as nrarr infin

For any ε gt 0 and k with 1 le k le n minus 1 let Snminus1kε = x isin Snminus1(1) |

|xi| lt ε for i = 1 2 k Thenint

Snminus1(1)Snminus1kε

a2ix2i dσ

nminus1(x) le σnminus1(Snminus1(1) Snminus1kε )

Snminus1kε

a2ix2i dσ

nminus1(x) le ε2ksum

a2i +nsum

which imply

supaiisinA

Snminus1(1)

a2ix2i dσ

nminus1(x)

le (σnminus1(Snminus1(1) Snminus1kε ) + ε2) sup

aiisinA

infinsum

a2i + supaiisinA

infinsum

Since σnminus1(Snminus1(1) Snminus1kε ) tends to zero as n rarr infin we obtain (2)

Proposition 42 Let aij i = 1 2 n(j) j = 1 2 be a se-

quence of positive real numbers where n(j) j = 1 2 is a se-

quence of positive integers divergent to infinity Let ai i = 1 2 be an l2-sequence of nonnegative real numbers We assume

limjrarrinfin

n(j)sum

(aij minus ai)2 = 0

Then we have

(1) ǫn(j)

converges weakly to γinfina2ii as j rarr infin

(2) σn(j)minus1

converges to γinfina2i iin the 2-Wasserstein metric as

j rarr infin

In particular En(j)

box converges to Γinfina2i i

as j rarr infin

Proof We first prove (2) We set aij = 0 for i ge n(j) + 1 Note thatthe assumption implies the l2-convergence of aij to ai as j rarr infinLemma 41(2) implies

lim supjrarrinfin

W2(σn(j)minus1

γn(j)

a2iji)

leradic2eminus1 lim sup

jrarrinfin

infinsum

a2ij =radic2eminus1

infinsum

a2i minusrarr 0 as k rarr infin

Gelbrichrsquos formula [7] tells us that

W2(γn(j)

a2iji γinfina2i

)2 =infinsum

(aij minus ai)2 minusrarr 0 as j rarr infin

By a triangle inequality we obtain (2)(1) is proved in the same way by using Lemma 41(1) and by remark-

ing dP2 leW2 This completes the proof

Lemma 43 Let bij i = 1 2 n(j) j = 1 2 be a sequence of

positive real numbers where n(j) j = 1 2 is a sequence of pos-

itive integers divergent to infinity Ifsumn(j)

i=1 b2ij converges to a positive

real number as j rarr infin then en(j)radicn(j)minus1 bij

has no subsequence con-

verging weakly to the Dirac measure δo at the origin o in H where we

embed the (solid) ellipsoids En(j)

radicn(j)minus1 bij

sub Rn(j) into the Hilbert space

H naturally and consider en(j)

radicn(j)minus1 biji

as Borel probability measures

Proof We first prove the lemma for σn(j)minus1

radicn(j)minus1 bij

It holds that

W2(σn(j)minus1

δo)2 =

Sn(j)minus1

n(j)minus1 biji

y2 dσn(j)minus1

Snminus1(1)

n(j)sum

(n(j)minus 1)b2ijx2i dσ

n(j)minus1(x)

n(j)sum

(n(j)minus 1)

Snminus1(1)

x21 dσn(j)minus1(x)

It follows from the Maxwell-Boltzmann distribution law that

limjrarrinfin

(n(j)minus 1)

Snminus1(1)

x21 dσn(j)minus1(x) = 1

We therefore have

limjrarrinfin

W2(σn(j)minus1

δo)2 = lim

jrarrinfin

n(j)sum

b2ij gt 0

so that σn(j)minus1

has no subsequence converging weakly to δo

We prove the lemma for ǫn(j)

radicn(j)minus1 bij

Applying Lemma 41(1) yields

thatlimjrarrinfin

dP(ǫn(j)

radicn(j)minus1 bij

γn(j)

b2ij) = 0

We also have

limjrarrinfin

W2(γn(j)

b2ij δo)

2 = limjrarrinfin

x2 dγn(j)b2ij(x) = lim

jrarrinfin

n(j)sum

b2ij gt 0

and hence γn(j)b2ij does not have a subsequence converging weakly to δo

and so does ǫn(j)radicn(j)minus1 bij

This completes the proof of the lemma

The following lemma is the special case of Theorem 12 where thelimit is a one-point mm-space

Lemma 44 Let aij i = 1 2 n(j) j = 1 2 be a sequence

of positive real numbers where n(j) j = 1 2 is a sequence of

positive integers divergent to infinity We assume that

limjrarrinfin

aij = 0 for any i(i)

lim infjrarrinfin

n(j)sum

a2ij gt 0(ii)

Then there exists no box convergent subsequence of En(j)

Proof Let aij be a sequence as in the assumption of the theoremSorting aij in ascending order in i we may assume that aij is mono-

tone nonincreasing in i for each j We suppose that En(j)

has a box convergent subsequence for which we use the same nota-

tion Then by (i) and Theorem 11 the box limit of En(j)

is mm-isomorphic to a one-point mm-space We set

n(j)sum

and bij =aij

maxAj 1

Since bij le aij we see that En(j)

is dominated by En(j)

so that En(j)

box converges to a one-point mm-space as j rarr

infin We remark that

lim infjrarrinfin

n(j)sum

b2ij gt 0 and

n(j)sum

b2ij le 1

Taking a subsequence again we assume thatsumn(j)

i=1 b2ij converges to a

positive real number as j rarr infin Applying Lemma 43 yields that

en(j)radicn(j)minus1 bij

has no subsequence converging weakly to δo in H Since

box converges to a one-point mm-space say lowast as j rarr infin

Lemma 213 implies that there is a sequence of εj-mm-isomorphisms

fj En(j)

rarr lowast with εj rarr 0+ as j rarr infin A nonexceptional

domain of fj has en(j)

radicn(j)minus1 bij

-measure at least 1minus εj and diameter at

most εj There is a closed metric ball Bj sub H of radius εj that contains

the nonexceptional domain of fj Note that en(j)

(Bj) ge 1 minusεj rarr 1 as j rarr infin If Bj were to contain the origin o of H for infinitely

many j then a subsequence of en(j)radicn(j)minus1 biji

would converge weakly

to δo which is a contradiction Thus all but finitely many Bj do notcontain the origin of H and Bj do not intersect minusBj for any such Bj

Since en(j)

is centrally symmetric with respect to the origin

we see that en(j)

(minusBj) = en(j)

(Bj) ge 1minus εj which is

a contradiction if j is large enough This completes the proof

Lemma 45 Let Xn n = 1 2 be a box convergent sequence

of mm-spaces and Yn n = 1 2 a sequence of mm-spaces with

Yn ≺ Xn Then Yn has a box convergent subsequence

Proof The lemma follows from [25 Lemma 428]

Proof of Theorem 12 We assume (A0)ndash(A3)The lsquoifrsquo part follows from Proposition 42

We prove the lsquoonly ifrsquo part Suppose that En(j)

is box

convergent and that aiji does not l2-converge to ai as j rarr infinWe first prove that ai is an l2-sequence This is because if not

then by Theorem 11 the weak limit of En(j)

is not an

mm-space which is a contradiction to the box convergence Replacingaiji with a subsequence with respect to the index j we assume that

limjrarrinfinsumn(j)

i=1 a2ij exists in [ 0+infin ] We prove

(43) limjrarrinfin

n(j)sum

a2ij gt

infinsum

In fact if the left-hand side of (43) is infinity then this is clear Ifnot the Banach-Alaoglu theorem tells us the existence of an l2-weaklyconvergent subsequence of aij Since aiji does not converge to ail2-strongly as j rarr infin we obtain (43)

Take a real number ε0 in such a way that

0 lt ε0 lt limjrarrinfin

n(j)sum

a2ij minusinfinsum

Setting

aijk =

akj if i le k

aij if i ge k + 1

we have

limjrarrinfin

n(j)sum

a2ijk = limjrarrinfin

a2ijk +

n(j)sum

= ka2k + limjrarrinfin

n(j)sum

a2ij minusksum

a2i gt ε0

Thus for any positive integer k there is j(k) such that

n(j(k))sum

a2ij(k)k gt ε0 and | akj(k) minus ak | lt1

Letting bik = aij(k)k we observe the following

bull bik le aij(k) for any i and kbull bik is monotone nonincreasing in i for each kbull b1k = a1j(k)k = akj(k) lt ak + 1k rarr 0 as k rarr infin

bullsumn(j(k))

i=1 b2ik gt ε0 gt 0 for any k

Consider Ek = En(j(k))

radicn(j(k))minus1 biki

It follows from Lemma 31 that Ek

is dominated by En(j(k))

radicn(j(k))minus1 aij(k)i

for any k and so Lemma 45 im-

plies that Ek has a box convergent subsequence However Lemma44 proves that Ek has no box convergent subsequence which is acontradiction This completes the proof

References

[1] V I Bogachev Gaussian measures Mathematical Surveys and Monographsvol 62 American Mathematical Society Providence RI 1998

[2] J Cheeger and T H Colding On the structure of spaces with Ricci curvature

bounded below I J Differential Geom 46 (1997) no 3 406ndash480[3] On the structure of spaces with Ricci curvature bounded below II J

Differential Geom 54 (2000) no 1 13ndash35[4] On the structure of spaces with Ricci curvature bounded below III J

Differential Geom 54 (2000) no 1 37ndash74

[5] K Fukaya Collapsing of Riemannian manifolds and eigenvalues of Laplace

operator Invent Math 87 (1987) no 3 517ndash547 DOI 101007BF01389241[6] K Funano and T Shioya Concentration Ricci curvature and eigenvalues of

Laplacian Geom Funct Anal 23 (2013) no 3 888ndash936 DOI 101007s00039-013-0215-x

[7] M Gelbrich On a formula for the L2 Wasserstein metric between measures

on Euclidean and Hilbert spaces Math Nachr 147 (1990) 185ndash203 DOI101002mana19901470121

[8] N Gigli A Mondino and G Savare Convergence of pointed non-compact met-

ric measure spaces and stability of Ricci curvature bounds and heat flows ProcLond Math Soc (3) 111 (2015) no 5 1071ndash1129 DOI 101112plmspdv047

[9] M Gromov Metric structures for Riemannian and non-Riemannian spacesReprint of the 2001 English edition Modern Birkhauser Classics BirkhauserBoston Inc Boston MA 2007 Based on the 1981 French original With ap-pendices by M Katz P Pansu and S Semmes Translated from the French bySean Michael Bates

[10] D Kazukawa Concentration of product spaces (2019) available atarXiv190911910[mathMG]

[11] D Kazukawa R Ozawa and N Suzuki Stabilities of rough curva-

ture dimension condition J Math Soc Japan posted on 2019 DOI102969jmsj81468146

[12] M Ledoux The concentration of measure phenomenon Mathematical Surveysand Monographs vol 89 American Mathematical Society Providence RI2001

[13] P Levy Problemes concrets drsquoanalyse fonctionnelle Avec un complement sur

les fonctionnelles analytiques par F Pellegrino Gauthier-Villars Paris 1951(French) 2d ed

[14] J Lott and C Villani Ricci curvature for metric-measure spaces via optimal

transport Ann of Math (2) 169 (2009) no 3 903ndash991 DOI 104007an-nals2009169903

[15] V D Milman The heritage of P Levy in geometrical functional analysisAsterisque 157-158 (1988) 273ndash301 Colloque Paul Levy sur les ProcessusStochastiques (Palaiseau 1987)

[16] H Nakajima Isoperimetric inequality on a metric measure space and Lipschitz

order with an additive error (2019) available at arXiv190207424[mathMG][17] R Ozawa Concentration function for pyramid and quantum metric mea-

sure space Proc Amer Math Soc 145 (2017) no 3 1301ndash1315 DOI101090proc13282

[18] R Ozawa and N Suzuki Stability of Talagrandrsquos inequality under concentra-

tion topology Proc Amer Math Soc 145 (2017) no 10 4493ndash4501 DOI101090proc13580

[19] R Ozawa and T Shioya Limit formulas for metric measure invariants

and phase transition property Math Z 280 (2015) no 3-4 759ndash782 DOI101007s00209-015-1447-2

[20] R Ozawa and T Yokota Stability of RCD condition under concentration topol-

ogy Calc Var Partial Differential Equations 58 (2019) no 4 Art 151 30DOI 101007s00526-019-1586-0

[21] G Perelman Ricci flow with surgery on three-manifolds (2003) available atarXivmath0303109[mathDG]

[22] V Pestov Dynamics of infinite-dimensional groups University Lecture Se-ries vol 40 American Mathematical Society Providence RI 2006 TheRamsey-Dvoretzky-Milman phenomenon Revised edition of Dynamics of

infinite-dimensional groups and Ramsey-type phenomena [Inst Mat Pura Apl(IMPA) Rio de Janeiro 2005 MR2164572]

[23] K-T Sturm On the geometry of metric measure spaces I Acta Math 196(2006) no 1 65ndash131 DOI 101007s11511-006-0002-8

[24] On the geometry of metric measure spaces II Acta Math 196 (2006)no 1 133ndash177 DOI 101007s11511-006-0003-7

[25] T ShioyaMetric measure geometry IRMA Lectures in Mathematics and The-oretical Physics vol 25 EMS Publishing House Zurich 2016 Gromovrsquos theoryof convergence and concentration of metrics and measures

[26] Metric measure limits of spheres and complex projective spaces Mea-sure theory in non-smooth spaces Partial Differ Equ Meas Theory DeGruyter Open Warsaw 2017 pp 261ndash287

[27] T Shioya and A Takatsu High-dimensional metric-measure limit of Stiefel

and flag manifolds Math Z 288 (2018) 1ndash35 DOI 101007s00209-018-2044-y

[28] T Shioya and T Yamaguchi Volume collapsed three-manifolds with a lower

curvature bound Math Ann 333 (2005) no 1 131ndash155 DOI 101007s00208-005-0667-x

Mathematical Institute Tohoku University Sendai 980-8578 JAPAN

E-mail address shioyamathtohokuacjpE-mail address daisukekazukawas6dctohokuacjp

1 Introduction

2 Preliminaries



23 Observable diameter


25 Pyramid

26 Dissipation

27 Asymptotic concentration

28 Gaussian space



References

distance functions the observable distance function dconc and the box

distance function on the set say X of isomorphism classes of metricmeasure spaces The box distance function is nearly a metrization ofmeasured Gromov-Hausdorff convergence (precisely the isomorphismclasses are little different) while the observable distance function in-duces a very characteristic topology called the concentration topologywhich is effective to capture the high-dimensional aspects of spacesThe concentration topology is weaker than the box topology and inparticular a measured Gromov-Hausdorff convergence becomes a con-vergence in the concentration topology He also introduced a naturalcompactification say Π of X with respect to the concentration topol-ogy where the topology on Π is called the weak topology The concen-tration topology is sometimes useful to investigate the dimension-freeproperties of manifolds For example it has been applied to obtain anew dimension-free estimate of eigenvalue ratios of the drifted Lapla-cian on a closed Riemannian manifold with nonnegative Bakry-EmeryRicci curvature [6]

The study of the concentration and weak topologies has been growingrapidly in recent years (see [6101116ndash202225ndash27]) However thereare only a few nontrivial examples of convergent sequences of met-ric measure spaces in the concentration and weak topologies wherelsquonontrivialrsquo means neither to be a Levy family (ie convergent to aone-point space) to infinitely dissipate (see Subsection 26 for dissi-pation) nor to be box convergent One way to construct a nontriv-ial convergent sequence is to take the disjoint union or the product(more generally the fibration) of trivial sequences and to perform lit-tle surgery on it (and also to repeat these procedures finitely manytimes) We call a sequence obtained in this way a reducible sequenceAn irreducible sequence is a sequence that is not reducible In thispaper any sequence of (solid) ellipsoids has a subsequence convergingto an infinite-dimensional Gaussian space in the concentrationweaktopology This provides a new family of nontrivial weak convergentsequences and especially contains the first discovered example of anirreducible nontrivial sequence that is convergent in the concentrationtopology

Let us state our main results precisely A solid ellipsoid and anellipsoid are respectively written as

Enαi = x isin Rn |nsum

x2iα2i

Snminus1αi = x isin Rn |

x2iα2i

for some se-

converges weakly to

infinsum

n(j)sum

(aij minus ai)2 = 0

En = En(j)

and Γ = Γinfina2i

of En of the domain

2 Preliminaries

dTV(micro ν) =1

∣∣∣∣dmicro

dωminus dν

∣∣∣∣ dω

where dmicrodω

XtimesXdX(x x

(le +infin)

lim supnrarrinfin

X = supp microX

X to Y

X and Y by

25 Pyramid

alent to each other

trating to P

γna2i =

notimes

γ1a2i

where a2i are

We call PΓinfina2i

γinfina2i =

infinotimes

γ1a2i

(21)infinsum

a2i lt +infin

nminus1 of

αi σnminus1αi)

dominated by Enβi

converges weakly

in Subsection 28

lim infjrarrinfin

ObsDiam(En(j)

minusκ)

ObsDiam(En(j)

minusκ)

ge limjrarrinfin

1minus κ)

R = (nminus 1)12

Inminus1

tnminus1eminust2

int infin

tmeminust2

Elowastγ

ϕS(x) =

radicnminus 1

rx x isin Rn o

radicnminus1ai

radicn

with N le n

dϕEx le

radic1 + CNε

θand dϕS

x leradic1 + CNε

dϕEx(v)2 =

)partr

partxj〈x v〉+ R

〈x v〉2nsum

x2ja4j

r2part

)〈x v〉

vjxja2j

x2 = 1 +Nminus1sum

a2jminus 1

x2 = 1 +O(Nε2)

and so

x = 1 +O(Nε2)xar

= 1 +O(Nε2)

We also have

x2nsum

x2ja4j

= 1 +Nminus1sum

a4jminus 1

x2 = 1 + O(Nε2)

vjxja2j

vjxjx +

Nminus1sum

a2jminus 1

)vjxjx =

〈x v〉x +O(Nε)

) we have

dϕEx(v)2 = t2g2(1 +O(Nε2)) +

r(1 +O(Nε2))(31)

rO(Nε) +

2 dt we have

partr=

minus 1n

nand r ge θ

radicn yields

0 le partR

partrle

radicn

partrminus R

∣∣∣∣ le1

dϕEx(v)2 = t2g2 +

r2+O(θminus2Nε)

= t2(partR

+ (1minus t2)R2

r2+O(θminus2Nε)

limnrarrinfin

enradicnminus1ai(D

nNε) = 1(1)

limnrarrinfin

γna2i (Dn

nNε) = σnminus1

radicnminus1 ai

(DnNε) =

γna2i (Dn

ϕE if (En

radicnminus1ai e

nradicnminus1ai)

radicnminus1 ai σ

NεcapFn

|DnNε

capFnθ) and so

radicnminus1ai|Dn

Nε en

|DnNε

capFnθ γna2i

) rarr 0

PΓinfina2i

as nrarr infin

radicn(j)minus1 ai

Since En(j)

radicn(j)minus1 ai

radicnminus1 ai for

dropping

i dominates Γka2i

and so PΓinfina2i

supPΓinfin

a2i We thus obtain

which is a contra-

converges

The ellipsoid En(j)

converges

bεi =

ai if i le I(ε)

n(j)aiji ≺ E

n(j)(1+2ε)bεi = (1 + 2ε)E

PEn(j)

bεi =

2ε if i ge I(ε)

En(j)aiji ≺ E

in PΓinfinb2εi

I(ε)minus1

(1+ε)2a2i

and ΓkminusI(ε)+1

ρ(PΓkb2εiPΓ

I(ε)minus1

kb2εi

ΓI(ε)minus1

(1+ε)2a2i ) le τ(ε)

ρ(PΓinfinb2εi

PΓI(ε)minus1

(1+ε)2a2i) le τ(ε)

b2ε(l)i

I(ε(l))minus1

infin as

as l rarr infin

PΓinfina2i

does not converge

PEn(j)

i as j rarr infin

asymptotically

a2 is not

i contains PΓinfin

i is not

a2ij we have

ObsDiam(Γn(j)

a2ijconverges

a2i as j rarr

converges

so that

and so PΓn(j)

a2ijto PΓinfin

converges

such that

supaiisinA

infinsum

a2i lt +infin

Then we have

limnrarrinfin

supaiisinA

) = 0(1)

lim supnrarrinfin

supaiisinA

aiisinA

infinsum

a2i(2)

radicnminus1

radicnminus1 ])

radicnminus 1 )

We remark that

radicnminus 1 ]))

limnrarrinfin

wθn = 1

It then holds that

nθ ) le dTV(ǫ

nradicnminus1 ai ǫ

dP(γna2i

ψ(x) =R

radicnminus 1 ])

W2(ǫnθ γ

2 leint

rminus 1

x2 dγnθ (x)

radicnminus 1 ]))

x2 dγna2i (x)

and hence

lim supnrarrinfin

supaiisinA

aiisinA

infinsum

This proves (1)

radicnminus1ai

)2 leint

Snminus1(1)

a2ix2i dσ

nminus1(x) middot 1

Inminus1

int infin

Inminus1

int infin

22 dr le 1

Inminus1

simradic2eminus12

a2ix2i dσ

Snminus1kε

a2ix2i dσ

a2i +nsum

which imply

supaiisinA

Snminus1(1)

a2ix2i dσ

nminus1(x)

aiisinA

infinsum

a2i + supaiisinA

infinsum

limjrarrinfin

n(j)sum

(aij minus ai)2 = 0

Then we have

(1) ǫn(j)

(2) σn(j)minus1

j rarr infin

In particular En(j)

as j rarr infin

lim supjrarrinfin

W2(σn(j)minus1

γn(j)

a2iji)

jrarrinfin

infinsum

a2ij =radic2eminus1

infinsum

W2(γn(j)

a2iji γinfina2i

)2 =infinsum

radicn(j)minus1 bij

It holds that

W2(σn(j)minus1

δo)2 =

Sn(j)minus1

n(j)minus1 biji

y2 dσn(j)minus1

Snminus1(1)

n(j)sum

n(j)minus1(x)

n(j)sum

(n(j)minus 1)

Snminus1(1)

limjrarrinfin

(n(j)minus 1)

Snminus1(1)

We therefore have

limjrarrinfin

W2(σn(j)minus1

δo)2 = lim

jrarrinfin

n(j)sum

b2ij gt 0

radicn(j)minus1 bij

thatlimjrarrinfin

dP(ǫn(j)

radicn(j)minus1 bij

γn(j)

b2ij) = 0

We also have

limjrarrinfin

W2(γn(j)

b2ij δo)

2 = limjrarrinfin

jrarrinfin

n(j)sum

b2ij gt 0

limjrarrinfin

lim infjrarrinfin

n(j)sum

a2ij gt 0(ii)

n(j)sum

and bij =aij

maxAj 1

so that En(j)

lim infjrarrinfin

n(j)sum

b2ij gt 0 and

n(j)sum

b2ij le 1

fj En(j)

radicn(j)minus1 bij

Since en(j)

we see that en(j)

(minusBj) = en(j)

is box

is not an

(43) limjrarrinfin

n(j)sum

a2ij gt

infinsum

n(j)sum

a2ij minusinfinsum

Setting

aijk =

akj if i le k

aij if i ge k + 1

we have

limjrarrinfin

n(j)sum

a2ijk +

n(j)sum

a2ij minusksum

a2i gt ε0

n(j(k))sum

bullsumn(j(k))

References

1 Introduction

2 Preliminaries





25 Pyramid

26 Dissipation


28 Gaussian space



References

for some se-

converges weakly to

infinsum

n(j)sum

(aij minus ai)2 = 0

En = En(j)

and Γ = Γinfina2i

of En of the domain

2 Preliminaries

dTV(micro ν) =1

∣∣∣∣dmicro

dωminus dν

∣∣∣∣ dω

where dmicrodω

XtimesXdX(x x

(le +infin)

lim supnrarrinfin

X = supp microX

X to Y

X and Y by

25 Pyramid

alent to each other

trating to P

γna2i =

notimes

γ1a2i

where a2i are

We call PΓinfina2i

γinfina2i =

infinotimes

γ1a2i

(21)infinsum

a2i lt +infin

nminus1 of

αi σnminus1αi)

dominated by Enβi

converges weakly

in Subsection 28

lim infjrarrinfin

ObsDiam(En(j)

minusκ)

ObsDiam(En(j)

minusκ)

ge limjrarrinfin

1minus κ)

R = (nminus 1)12

Inminus1

tnminus1eminust2

int infin

tmeminust2

Elowastγ

ϕS(x) =

radicnminus 1

rx x isin Rn o

radicnminus1ai

radicn

with N le n

dϕEx le

radic1 + CNε

θand dϕS

x leradic1 + CNε

dϕEx(v)2 =

)partr

partxj〈x v〉+ R

〈x v〉2nsum

x2ja4j

r2part

)〈x v〉

vjxja2j

x2 = 1 +Nminus1sum

a2jminus 1

x2 = 1 +O(Nε2)

and so

x = 1 +O(Nε2)xar

= 1 +O(Nε2)

We also have

x2nsum

x2ja4j

= 1 +Nminus1sum

a4jminus 1

x2 = 1 + O(Nε2)

vjxja2j

vjxjx +

Nminus1sum

a2jminus 1

)vjxjx =

〈x v〉x +O(Nε)

) we have

dϕEx(v)2 = t2g2(1 +O(Nε2)) +

r(1 +O(Nε2))(31)

rO(Nε) +

2 dt we have

partr=

minus 1n

nand r ge θ

radicn yields

0 le partR

partrle

radicn

partrminus R

∣∣∣∣ le1

dϕEx(v)2 = t2g2 +

r2+O(θminus2Nε)

= t2(partR

+ (1minus t2)R2

r2+O(θminus2Nε)

limnrarrinfin

enradicnminus1ai(D

nNε) = 1(1)

limnrarrinfin

γna2i (Dn

nNε) = σnminus1

radicnminus1 ai

(DnNε) =

γna2i (Dn

ϕE if (En

radicnminus1ai e

nradicnminus1ai)

radicnminus1 ai σ

NεcapFn

|DnNε

capFnθ) and so

radicnminus1ai|Dn

Nε en

|DnNε

capFnθ γna2i

) rarr 0

PΓinfina2i

as nrarr infin

radicn(j)minus1 ai

Since En(j)

radicn(j)minus1 ai

radicnminus1 ai for

dropping

i dominates Γka2i

and so PΓinfina2i

supPΓinfin

a2i We thus obtain

which is a contra-

converges

The ellipsoid En(j)

converges

bεi =

ai if i le I(ε)

n(j)aiji ≺ E

n(j)(1+2ε)bεi = (1 + 2ε)E

PEn(j)

bεi =

2ε if i ge I(ε)

En(j)aiji ≺ E

in PΓinfinb2εi

I(ε)minus1

(1+ε)2a2i

and ΓkminusI(ε)+1

ρ(PΓkb2εiPΓ

I(ε)minus1

kb2εi

ΓI(ε)minus1

(1+ε)2a2i ) le τ(ε)

ρ(PΓinfinb2εi

PΓI(ε)minus1

(1+ε)2a2i) le τ(ε)

b2ε(l)i

I(ε(l))minus1

infin as

as l rarr infin

PΓinfina2i

does not converge

PEn(j)

i as j rarr infin

asymptotically

a2 is not

i contains PΓinfin

i is not

a2ij we have

ObsDiam(Γn(j)

a2ijconverges

a2i as j rarr

converges

so that

and so PΓn(j)

a2ijto PΓinfin

converges

such that

supaiisinA

infinsum

a2i lt +infin

Then we have

limnrarrinfin

supaiisinA

) = 0(1)

lim supnrarrinfin

supaiisinA

aiisinA

infinsum

a2i(2)

radicnminus1

radicnminus1 ])

radicnminus 1 )

We remark that

radicnminus 1 ]))

limnrarrinfin

wθn = 1

It then holds that

nθ ) le dTV(ǫ

nradicnminus1 ai ǫ

dP(γna2i

ψ(x) =R

radicnminus 1 ])

W2(ǫnθ γ

2 leint

rminus 1

x2 dγnθ (x)

radicnminus 1 ]))

x2 dγna2i (x)

and hence

lim supnrarrinfin

supaiisinA

aiisinA

infinsum

This proves (1)

radicnminus1ai

)2 leint

Snminus1(1)

a2ix2i dσ

nminus1(x) middot 1

Inminus1

int infin

Inminus1

int infin

22 dr le 1

Inminus1

simradic2eminus12

a2ix2i dσ

Snminus1kε

a2ix2i dσ

a2i +nsum

which imply

supaiisinA

Snminus1(1)

a2ix2i dσ

nminus1(x)

aiisinA

infinsum

a2i + supaiisinA

infinsum

limjrarrinfin

n(j)sum

(aij minus ai)2 = 0

Then we have

(1) ǫn(j)

(2) σn(j)minus1

j rarr infin

In particular En(j)

as j rarr infin

lim supjrarrinfin

W2(σn(j)minus1

γn(j)

a2iji)

jrarrinfin

infinsum

a2ij =radic2eminus1

infinsum

W2(γn(j)

a2iji γinfina2i

)2 =infinsum

radicn(j)minus1 bij

It holds that

W2(σn(j)minus1

δo)2 =

Sn(j)minus1

n(j)minus1 biji

y2 dσn(j)minus1

Snminus1(1)

n(j)sum

n(j)minus1(x)

n(j)sum

(n(j)minus 1)

Snminus1(1)

limjrarrinfin

(n(j)minus 1)

Snminus1(1)

We therefore have

limjrarrinfin

W2(σn(j)minus1

δo)2 = lim

jrarrinfin

n(j)sum

b2ij gt 0

radicn(j)minus1 bij

thatlimjrarrinfin

dP(ǫn(j)

radicn(j)minus1 bij

γn(j)

b2ij) = 0

We also have

limjrarrinfin

W2(γn(j)

b2ij δo)

2 = limjrarrinfin

jrarrinfin

n(j)sum

b2ij gt 0

limjrarrinfin

lim infjrarrinfin

n(j)sum

a2ij gt 0(ii)

n(j)sum

and bij =aij

maxAj 1

so that En(j)

lim infjrarrinfin

n(j)sum

b2ij gt 0 and

n(j)sum

b2ij le 1

fj En(j)

radicn(j)minus1 bij

Since en(j)

we see that en(j)

(minusBj) = en(j)

is box

is not an

(43) limjrarrinfin

n(j)sum

a2ij gt

infinsum

n(j)sum

a2ij minusinfinsum

Setting

aijk =

akj if i le k

aij if i ge k + 1

we have

limjrarrinfin

n(j)sum

a2ijk +

n(j)sum

a2ij minusksum

a2i gt ε0

n(j(k))sum

bullsumn(j(k))

References

1 Introduction

2 Preliminaries





25 Pyramid

26 Dissipation


28 Gaussian space



References

converges weakly to

infinsum

n(j)sum

(aij minus ai)2 = 0

En = En(j)

and Γ = Γinfina2i

of En of the domain

2 Preliminaries

dTV(micro ν) =1

∣∣∣∣dmicro

dωminus dν

∣∣∣∣ dω

where dmicrodω

XtimesXdX(x x

(le +infin)

lim supnrarrinfin

X = supp microX

X to Y

X and Y by

25 Pyramid

alent to each other

trating to P

γna2i =

notimes

γ1a2i

where a2i are

We call PΓinfina2i

γinfina2i =

infinotimes

γ1a2i

(21)infinsum

a2i lt +infin

nminus1 of

αi σnminus1αi)

dominated by Enβi

converges weakly

in Subsection 28

lim infjrarrinfin

ObsDiam(En(j)

minusκ)

ObsDiam(En(j)

minusκ)

ge limjrarrinfin

1minus κ)

R = (nminus 1)12

Inminus1

tnminus1eminust2

int infin

tmeminust2

Elowastγ

ϕS(x) =

radicnminus 1

rx x isin Rn o

radicnminus1ai

radicn

with N le n

dϕEx le

radic1 + CNε

θand dϕS

x leradic1 + CNε

dϕEx(v)2 =

)partr

partxj〈x v〉+ R

〈x v〉2nsum

x2ja4j

r2part

)〈x v〉

vjxja2j

x2 = 1 +Nminus1sum

a2jminus 1

x2 = 1 +O(Nε2)

and so

x = 1 +O(Nε2)xar

= 1 +O(Nε2)

We also have

x2nsum

x2ja4j

= 1 +Nminus1sum

a4jminus 1

x2 = 1 + O(Nε2)

vjxja2j

vjxjx +

Nminus1sum

a2jminus 1

)vjxjx =

〈x v〉x +O(Nε)

) we have

dϕEx(v)2 = t2g2(1 +O(Nε2)) +

r(1 +O(Nε2))(31)

rO(Nε) +

2 dt we have

partr=

minus 1n

nand r ge θ

radicn yields

0 le partR

partrle

radicn

partrminus R

∣∣∣∣ le1

dϕEx(v)2 = t2g2 +

r2+O(θminus2Nε)

= t2(partR

+ (1minus t2)R2

r2+O(θminus2Nε)

limnrarrinfin

enradicnminus1ai(D

nNε) = 1(1)

limnrarrinfin

γna2i (Dn

nNε) = σnminus1

radicnminus1 ai

(DnNε) =

γna2i (Dn

ϕE if (En

radicnminus1ai e

nradicnminus1ai)

radicnminus1 ai σ

NεcapFn

|DnNε

capFnθ) and so

radicnminus1ai|Dn

Nε en

|DnNε

capFnθ γna2i

) rarr 0

PΓinfina2i

as nrarr infin

radicn(j)minus1 ai

Since En(j)

radicn(j)minus1 ai

radicnminus1 ai for

dropping

i dominates Γka2i

and so PΓinfina2i

supPΓinfin

a2i We thus obtain

which is a contra-

converges

The ellipsoid En(j)

converges

bεi =

ai if i le I(ε)

n(j)aiji ≺ E

n(j)(1+2ε)bεi = (1 + 2ε)E

PEn(j)

bεi =

2ε if i ge I(ε)

En(j)aiji ≺ E

in PΓinfinb2εi

I(ε)minus1

(1+ε)2a2i

and ΓkminusI(ε)+1

ρ(PΓkb2εiPΓ

I(ε)minus1

kb2εi

ΓI(ε)minus1

(1+ε)2a2i ) le τ(ε)

ρ(PΓinfinb2εi

PΓI(ε)minus1

(1+ε)2a2i) le τ(ε)

b2ε(l)i

I(ε(l))minus1

infin as

as l rarr infin

PΓinfina2i

does not converge

PEn(j)

i as j rarr infin

asymptotically

a2 is not

i contains PΓinfin

i is not

a2ij we have

ObsDiam(Γn(j)

a2ijconverges

a2i as j rarr

converges

so that

and so PΓn(j)

a2ijto PΓinfin

converges

such that

supaiisinA

infinsum

a2i lt +infin

Then we have

limnrarrinfin

supaiisinA

) = 0(1)

lim supnrarrinfin

supaiisinA

aiisinA

infinsum

a2i(2)

radicnminus1

radicnminus1 ])

radicnminus 1 )

We remark that

radicnminus 1 ]))

limnrarrinfin

wθn = 1

It then holds that

nθ ) le dTV(ǫ

nradicnminus1 ai ǫ

dP(γna2i

ψ(x) =R

radicnminus 1 ])

W2(ǫnθ γ

2 leint

rminus 1

x2 dγnθ (x)

radicnminus 1 ]))

x2 dγna2i (x)

and hence

lim supnrarrinfin

supaiisinA

aiisinA

infinsum

This proves (1)

radicnminus1ai

)2 leint

Snminus1(1)

a2ix2i dσ

nminus1(x) middot 1

Inminus1

int infin

Inminus1

int infin

22 dr le 1

Inminus1

simradic2eminus12

a2ix2i dσ

Snminus1kε

a2ix2i dσ

a2i +nsum

which imply

supaiisinA

Snminus1(1)

a2ix2i dσ

nminus1(x)

aiisinA

infinsum

a2i + supaiisinA

infinsum

limjrarrinfin

n(j)sum

(aij minus ai)2 = 0

Then we have

(1) ǫn(j)

(2) σn(j)minus1

j rarr infin

In particular En(j)

as j rarr infin

lim supjrarrinfin

W2(σn(j)minus1

γn(j)

a2iji)

jrarrinfin

infinsum

a2ij =radic2eminus1

infinsum

W2(γn(j)

a2iji γinfina2i

)2 =infinsum

radicn(j)minus1 bij

It holds that

W2(σn(j)minus1

δo)2 =

Sn(j)minus1

n(j)minus1 biji

y2 dσn(j)minus1

Snminus1(1)

n(j)sum

n(j)minus1(x)

n(j)sum

(n(j)minus 1)

Snminus1(1)

limjrarrinfin

(n(j)minus 1)

Snminus1(1)

We therefore have

limjrarrinfin

W2(σn(j)minus1

δo)2 = lim

jrarrinfin

n(j)sum

b2ij gt 0

radicn(j)minus1 bij

thatlimjrarrinfin

dP(ǫn(j)

radicn(j)minus1 bij

γn(j)

b2ij) = 0

We also have

limjrarrinfin

W2(γn(j)

b2ij δo)

2 = limjrarrinfin

jrarrinfin

n(j)sum

b2ij gt 0

limjrarrinfin

lim infjrarrinfin

n(j)sum

a2ij gt 0(ii)

n(j)sum

and bij =aij

maxAj 1

so that En(j)

lim infjrarrinfin

n(j)sum

b2ij gt 0 and

n(j)sum

b2ij le 1

fj En(j)

radicn(j)minus1 bij

Since en(j)

we see that en(j)

(minusBj) = en(j)

is box

is not an

(43) limjrarrinfin

n(j)sum

a2ij gt

infinsum

n(j)sum

a2ij minusinfinsum

Setting

aijk =

akj if i le k

aij if i ge k + 1

we have

limjrarrinfin

n(j)sum

a2ijk +

n(j)sum

a2ij minusksum

a2i gt ε0

n(j(k))sum

bullsumn(j(k))

References

1 Introduction

2 Preliminaries





25 Pyramid

26 Dissipation


28 Gaussian space



References

En = En(j)

and Γ = Γinfina2i

of En of the domain

2 Preliminaries

dTV(micro ν) =1

∣∣∣∣dmicro

dωminus dν

∣∣∣∣ dω

where dmicrodω

XtimesXdX(x x

(le +infin)

lim supnrarrinfin

X = supp microX

X to Y

X and Y by

25 Pyramid

alent to each other

trating to P

γna2i =

notimes

γ1a2i

where a2i are

We call PΓinfina2i

γinfina2i =

infinotimes

γ1a2i

(21)infinsum

a2i lt +infin

nminus1 of

αi σnminus1αi)

dominated by Enβi

converges weakly

in Subsection 28

lim infjrarrinfin

ObsDiam(En(j)

minusκ)

ObsDiam(En(j)

minusκ)

ge limjrarrinfin

1minus κ)

R = (nminus 1)12

Inminus1

tnminus1eminust2

int infin

tmeminust2

Elowastγ

ϕS(x) =

radicnminus 1

rx x isin Rn o

radicnminus1ai

radicn

with N le n

dϕEx le

radic1 + CNε

θand dϕS

x leradic1 + CNε

dϕEx(v)2 =

)partr

partxj〈x v〉+ R

〈x v〉2nsum

x2ja4j

r2part

)〈x v〉

vjxja2j

x2 = 1 +Nminus1sum

a2jminus 1

x2 = 1 +O(Nε2)

and so

x = 1 +O(Nε2)xar

= 1 +O(Nε2)

We also have

x2nsum

x2ja4j

= 1 +Nminus1sum

a4jminus 1

x2 = 1 + O(Nε2)

vjxja2j

vjxjx +

Nminus1sum

a2jminus 1

)vjxjx =

〈x v〉x +O(Nε)

) we have

dϕEx(v)2 = t2g2(1 +O(Nε2)) +

r(1 +O(Nε2))(31)

rO(Nε) +

2 dt we have

partr=

minus 1n

nand r ge θ

radicn yields

0 le partR

partrle

radicn

partrminus R

∣∣∣∣ le1

dϕEx(v)2 = t2g2 +

r2+O(θminus2Nε)

= t2(partR

+ (1minus t2)R2

r2+O(θminus2Nε)

limnrarrinfin

enradicnminus1ai(D

nNε) = 1(1)

limnrarrinfin

γna2i (Dn

nNε) = σnminus1

radicnminus1 ai

(DnNε) =

γna2i (Dn

ϕE if (En

radicnminus1ai e

nradicnminus1ai)

radicnminus1 ai σ

NεcapFn

|DnNε

capFnθ) and so

radicnminus1ai|Dn

Nε en

|DnNε

capFnθ γna2i

) rarr 0

PΓinfina2i

as nrarr infin

radicn(j)minus1 ai

Since En(j)

radicn(j)minus1 ai

radicnminus1 ai for

dropping

i dominates Γka2i

and so PΓinfina2i

supPΓinfin

a2i We thus obtain

which is a contra-

converges

The ellipsoid En(j)

converges

bεi =

ai if i le I(ε)

n(j)aiji ≺ E

n(j)(1+2ε)bεi = (1 + 2ε)E

PEn(j)

bεi =

2ε if i ge I(ε)

En(j)aiji ≺ E

in PΓinfinb2εi

I(ε)minus1

(1+ε)2a2i

and ΓkminusI(ε)+1

ρ(PΓkb2εiPΓ

I(ε)minus1

kb2εi

ΓI(ε)minus1

(1+ε)2a2i ) le τ(ε)

ρ(PΓinfinb2εi

PΓI(ε)minus1

(1+ε)2a2i) le τ(ε)

b2ε(l)i

I(ε(l))minus1

infin as

as l rarr infin

PΓinfina2i

does not converge

PEn(j)

i as j rarr infin

asymptotically

a2 is not

i contains PΓinfin

i is not

a2ij we have

ObsDiam(Γn(j)

a2ijconverges

a2i as j rarr

converges

so that

and so PΓn(j)

a2ijto PΓinfin

converges

such that

supaiisinA

infinsum

a2i lt +infin

Then we have

limnrarrinfin

supaiisinA

) = 0(1)

lim supnrarrinfin

supaiisinA

aiisinA

infinsum

a2i(2)

radicnminus1

radicnminus1 ])

radicnminus 1 )

We remark that

radicnminus 1 ]))

limnrarrinfin

wθn = 1

It then holds that

nθ ) le dTV(ǫ

nradicnminus1 ai ǫ

dP(γna2i

ψ(x) =R

radicnminus 1 ])

W2(ǫnθ γ

2 leint

rminus 1

x2 dγnθ (x)

radicnminus 1 ]))

x2 dγna2i (x)

and hence

lim supnrarrinfin

supaiisinA

aiisinA

infinsum

This proves (1)

radicnminus1ai

)2 leint

Snminus1(1)

a2ix2i dσ

nminus1(x) middot 1

Inminus1

int infin

Inminus1

int infin

22 dr le 1

Inminus1

simradic2eminus12

a2ix2i dσ

Snminus1kε

a2ix2i dσ

a2i +nsum

which imply

supaiisinA

Snminus1(1)

a2ix2i dσ

nminus1(x)

aiisinA

infinsum

a2i + supaiisinA

infinsum

limjrarrinfin

n(j)sum

(aij minus ai)2 = 0

Then we have

(1) ǫn(j)

(2) σn(j)minus1

j rarr infin

In particular En(j)

as j rarr infin

lim supjrarrinfin

W2(σn(j)minus1

γn(j)

a2iji)

jrarrinfin

infinsum

a2ij =radic2eminus1

infinsum

W2(γn(j)

a2iji γinfina2i

)2 =infinsum

radicn(j)minus1 bij

It holds that

W2(σn(j)minus1

δo)2 =

Sn(j)minus1

n(j)minus1 biji

y2 dσn(j)minus1

Snminus1(1)

n(j)sum

n(j)minus1(x)

n(j)sum

(n(j)minus 1)

Snminus1(1)

limjrarrinfin

(n(j)minus 1)

Snminus1(1)

We therefore have

limjrarrinfin

W2(σn(j)minus1

δo)2 = lim

jrarrinfin

n(j)sum

b2ij gt 0

radicn(j)minus1 bij

thatlimjrarrinfin

dP(ǫn(j)

radicn(j)minus1 bij

γn(j)

b2ij) = 0

We also have

limjrarrinfin

W2(γn(j)

b2ij δo)

2 = limjrarrinfin

jrarrinfin

n(j)sum

b2ij gt 0

limjrarrinfin

lim infjrarrinfin

n(j)sum

a2ij gt 0(ii)

n(j)sum

and bij =aij

maxAj 1

so that En(j)

lim infjrarrinfin

n(j)sum

b2ij gt 0 and

n(j)sum

b2ij le 1

fj En(j)

radicn(j)minus1 bij

Since en(j)

we see that en(j)

(minusBj) = en(j)

is box

is not an

(43) limjrarrinfin

n(j)sum

a2ij gt

infinsum

n(j)sum

a2ij minusinfinsum

Setting

aijk =

akj if i le k

aij if i ge k + 1

we have

limjrarrinfin

n(j)sum

a2ijk +

n(j)sum

a2ij minusksum

a2i gt ε0

n(j(k))sum

bullsumn(j(k))

References

1 Introduction

2 Preliminaries





25 Pyramid

26 Dissipation


28 Gaussian space



References

dTV(micro ν) =1

∣∣∣∣dmicro

dωminus dν

∣∣∣∣ dω

where dmicrodω

XtimesXdX(x x

(le +infin)

lim supnrarrinfin

X = supp microX

X to Y

X and Y by

25 Pyramid

alent to each other

trating to P

γna2i =

notimes

γ1a2i

where a2i are

We call PΓinfina2i

γinfina2i =

infinotimes

γ1a2i

(21)infinsum

a2i lt +infin

nminus1 of

αi σnminus1αi)

dominated by Enβi

converges weakly

in Subsection 28

lim infjrarrinfin

ObsDiam(En(j)

minusκ)

ObsDiam(En(j)

minusκ)

ge limjrarrinfin

1minus κ)

R = (nminus 1)12

Inminus1

tnminus1eminust2

int infin

tmeminust2

Elowastγ

ϕS(x) =

radicnminus 1

rx x isin Rn o

radicnminus1ai

radicn

with N le n

dϕEx le

radic1 + CNε

θand dϕS

x leradic1 + CNε

dϕEx(v)2 =

)partr

partxj〈x v〉+ R

〈x v〉2nsum

x2ja4j

r2part

)〈x v〉

vjxja2j

x2 = 1 +Nminus1sum

a2jminus 1

x2 = 1 +O(Nε2)

and so

x = 1 +O(Nε2)xar

= 1 +O(Nε2)

We also have

x2nsum

x2ja4j

= 1 +Nminus1sum

a4jminus 1

x2 = 1 + O(Nε2)

vjxja2j

vjxjx +

Nminus1sum

a2jminus 1

)vjxjx =

〈x v〉x +O(Nε)

) we have

dϕEx(v)2 = t2g2(1 +O(Nε2)) +

r(1 +O(Nε2))(31)

rO(Nε) +

2 dt we have

partr=

minus 1n

nand r ge θ

radicn yields

0 le partR

partrle

radicn

partrminus R

∣∣∣∣ le1

dϕEx(v)2 = t2g2 +

r2+O(θminus2Nε)

= t2(partR

+ (1minus t2)R2

r2+O(θminus2Nε)

limnrarrinfin

enradicnminus1ai(D

nNε) = 1(1)

limnrarrinfin

γna2i (Dn

nNε) = σnminus1

radicnminus1 ai

(DnNε) =

γna2i (Dn

ϕE if (En

radicnminus1ai e

nradicnminus1ai)

radicnminus1 ai σ

NεcapFn

|DnNε

capFnθ) and so

radicnminus1ai|Dn

Nε en

|DnNε

capFnθ γna2i

) rarr 0

PΓinfina2i

as nrarr infin

radicn(j)minus1 ai

Since En(j)

radicn(j)minus1 ai

radicnminus1 ai for

dropping

i dominates Γka2i

and so PΓinfina2i

supPΓinfin

a2i We thus obtain

which is a contra-

converges

The ellipsoid En(j)

converges

bεi =

ai if i le I(ε)

n(j)aiji ≺ E

n(j)(1+2ε)bεi = (1 + 2ε)E

PEn(j)

bεi =

2ε if i ge I(ε)

En(j)aiji ≺ E

in PΓinfinb2εi

I(ε)minus1

(1+ε)2a2i

and ΓkminusI(ε)+1

ρ(PΓkb2εiPΓ

I(ε)minus1

kb2εi

ΓI(ε)minus1

(1+ε)2a2i ) le τ(ε)

ρ(PΓinfinb2εi

PΓI(ε)minus1

(1+ε)2a2i) le τ(ε)

b2ε(l)i

I(ε(l))minus1

infin as

as l rarr infin

PΓinfina2i

does not converge

PEn(j)

i as j rarr infin

asymptotically

a2 is not

i contains PΓinfin

i is not

a2ij we have

ObsDiam(Γn(j)

a2ijconverges

a2i as j rarr

converges

so that

and so PΓn(j)

a2ijto PΓinfin

converges

such that

supaiisinA

infinsum

a2i lt +infin

Then we have

limnrarrinfin

supaiisinA

) = 0(1)

lim supnrarrinfin

supaiisinA

aiisinA

infinsum

a2i(2)

radicnminus1

radicnminus1 ])

radicnminus 1 )

We remark that

radicnminus 1 ]))

limnrarrinfin

wθn = 1

It then holds that

nθ ) le dTV(ǫ

nradicnminus1 ai ǫ

dP(γna2i

ψ(x) =R

radicnminus 1 ])

W2(ǫnθ γ

2 leint

rminus 1

x2 dγnθ (x)

radicnminus 1 ]))

x2 dγna2i (x)

and hence

lim supnrarrinfin

supaiisinA

aiisinA

infinsum

This proves (1)

radicnminus1ai

)2 leint

Snminus1(1)

a2ix2i dσ

nminus1(x) middot 1

Inminus1

int infin

Inminus1

int infin

22 dr le 1

Inminus1

simradic2eminus12

a2ix2i dσ

Snminus1kε

a2ix2i dσ

a2i +nsum

which imply

supaiisinA

Snminus1(1)

a2ix2i dσ

nminus1(x)

aiisinA

infinsum

a2i + supaiisinA

infinsum

limjrarrinfin

n(j)sum

(aij minus ai)2 = 0

Then we have

(1) ǫn(j)

(2) σn(j)minus1

j rarr infin

In particular En(j)

as j rarr infin

lim supjrarrinfin

W2(σn(j)minus1

γn(j)

a2iji)

jrarrinfin

infinsum

a2ij =radic2eminus1

infinsum

W2(γn(j)

a2iji γinfina2i

)2 =infinsum

radicn(j)minus1 bij

It holds that

W2(σn(j)minus1

δo)2 =

Sn(j)minus1

n(j)minus1 biji

y2 dσn(j)minus1

Snminus1(1)

n(j)sum

n(j)minus1(x)

n(j)sum

(n(j)minus 1)

Snminus1(1)

limjrarrinfin

(n(j)minus 1)

Snminus1(1)

We therefore have

limjrarrinfin

W2(σn(j)minus1

δo)2 = lim

jrarrinfin

n(j)sum

b2ij gt 0

radicn(j)minus1 bij

thatlimjrarrinfin

dP(ǫn(j)

radicn(j)minus1 bij

γn(j)

b2ij) = 0

We also have

limjrarrinfin

W2(γn(j)

b2ij δo)

2 = limjrarrinfin

jrarrinfin

n(j)sum

b2ij gt 0

limjrarrinfin

lim infjrarrinfin

n(j)sum

a2ij gt 0(ii)

n(j)sum

and bij =aij

maxAj 1

so that En(j)

lim infjrarrinfin

n(j)sum

b2ij gt 0 and

n(j)sum

b2ij le 1

fj En(j)

radicn(j)minus1 bij

Since en(j)

we see that en(j)

(minusBj) = en(j)

is box

is not an

(43) limjrarrinfin

n(j)sum

a2ij gt

infinsum

n(j)sum

a2ij minusinfinsum

Setting

aijk =

akj if i le k

aij if i ge k + 1

we have

limjrarrinfin

n(j)sum

a2ijk +

n(j)sum

a2ij minusksum

a2i gt ε0

n(j(k))sum

bullsumn(j(k))

References

1 Introduction

2 Preliminaries





25 Pyramid

26 Dissipation


28 Gaussian space



References

lim supnrarrinfin

X = supp microX

X to Y

X and Y by

25 Pyramid

alent to each other

trating to P

γna2i =

notimes

γ1a2i

where a2i are

We call PΓinfina2i

γinfina2i =

infinotimes

γ1a2i

(21)infinsum

a2i lt +infin

nminus1 of

αi σnminus1αi)

dominated by Enβi

converges weakly

in Subsection 28

lim infjrarrinfin

ObsDiam(En(j)

minusκ)

ObsDiam(En(j)

minusκ)

ge limjrarrinfin

1minus κ)

R = (nminus 1)12

Inminus1

tnminus1eminust2

int infin

tmeminust2

Elowastγ

ϕS(x) =

radicnminus 1

rx x isin Rn o

radicnminus1ai

radicn

with N le n

dϕEx le

radic1 + CNε

θand dϕS

x leradic1 + CNε

dϕEx(v)2 =

)partr

partxj〈x v〉+ R

〈x v〉2nsum

x2ja4j

r2part

)〈x v〉

vjxja2j

x2 = 1 +Nminus1sum

a2jminus 1

x2 = 1 +O(Nε2)

and so

x = 1 +O(Nε2)xar

= 1 +O(Nε2)

We also have

x2nsum

x2ja4j

= 1 +Nminus1sum

a4jminus 1

x2 = 1 + O(Nε2)

vjxja2j

vjxjx +

Nminus1sum

a2jminus 1

)vjxjx =

〈x v〉x +O(Nε)

) we have

dϕEx(v)2 = t2g2(1 +O(Nε2)) +

r(1 +O(Nε2))(31)

rO(Nε) +

2 dt we have

partr=

minus 1n

nand r ge θ

radicn yields

0 le partR

partrle

radicn

partrminus R

∣∣∣∣ le1

dϕEx(v)2 = t2g2 +

r2+O(θminus2Nε)

= t2(partR

+ (1minus t2)R2

r2+O(θminus2Nε)

limnrarrinfin

enradicnminus1ai(D

nNε) = 1(1)

limnrarrinfin

γna2i (Dn

nNε) = σnminus1

radicnminus1 ai

(DnNε) =

γna2i (Dn

ϕE if (En

radicnminus1ai e

nradicnminus1ai)

radicnminus1 ai σ

NεcapFn

|DnNε

capFnθ) and so

radicnminus1ai|Dn

Nε en

|DnNε

capFnθ γna2i

) rarr 0

PΓinfina2i

as nrarr infin

radicn(j)minus1 ai

Since En(j)

radicn(j)minus1 ai

radicnminus1 ai for

dropping

i dominates Γka2i

and so PΓinfina2i

supPΓinfin

a2i We thus obtain

which is a contra-

converges

The ellipsoid En(j)

converges

bεi =

ai if i le I(ε)

n(j)aiji ≺ E

n(j)(1+2ε)bεi = (1 + 2ε)E

PEn(j)

bεi =

2ε if i ge I(ε)

En(j)aiji ≺ E

in PΓinfinb2εi

I(ε)minus1

(1+ε)2a2i

and ΓkminusI(ε)+1

ρ(PΓkb2εiPΓ

I(ε)minus1

kb2εi

ΓI(ε)minus1

(1+ε)2a2i ) le τ(ε)

ρ(PΓinfinb2εi

PΓI(ε)minus1

(1+ε)2a2i) le τ(ε)

b2ε(l)i

I(ε(l))minus1

infin as

as l rarr infin

PΓinfina2i

does not converge

PEn(j)

i as j rarr infin

asymptotically

a2 is not

i contains PΓinfin

i is not

a2ij we have

ObsDiam(Γn(j)

a2ijconverges

a2i as j rarr

converges

so that

and so PΓn(j)

a2ijto PΓinfin

converges

such that

supaiisinA

infinsum

a2i lt +infin

Then we have

limnrarrinfin

supaiisinA

) = 0(1)

lim supnrarrinfin

supaiisinA

aiisinA

infinsum

a2i(2)

radicnminus1

radicnminus1 ])

radicnminus 1 )

We remark that

radicnminus 1 ]))

limnrarrinfin

wθn = 1

It then holds that

nθ ) le dTV(ǫ

nradicnminus1 ai ǫ

dP(γna2i

ψ(x) =R

radicnminus 1 ])

W2(ǫnθ γ

2 leint

rminus 1

x2 dγnθ (x)

radicnminus 1 ]))

x2 dγna2i (x)

and hence

lim supnrarrinfin

supaiisinA

aiisinA

infinsum

This proves (1)

radicnminus1ai

)2 leint

Snminus1(1)

a2ix2i dσ

nminus1(x) middot 1

Inminus1

int infin

Inminus1

int infin

22 dr le 1

Inminus1

simradic2eminus12

a2ix2i dσ

Snminus1kε

a2ix2i dσ

a2i +nsum

which imply

supaiisinA

Snminus1(1)

a2ix2i dσ

nminus1(x)

aiisinA

infinsum

a2i + supaiisinA

infinsum

limjrarrinfin

n(j)sum

(aij minus ai)2 = 0

Then we have

(1) ǫn(j)

(2) σn(j)minus1

j rarr infin

In particular En(j)

as j rarr infin

lim supjrarrinfin

W2(σn(j)minus1

γn(j)

a2iji)

jrarrinfin

infinsum

a2ij =radic2eminus1

infinsum

W2(γn(j)

a2iji γinfina2i

)2 =infinsum

radicn(j)minus1 bij

It holds that

W2(σn(j)minus1

δo)2 =

Sn(j)minus1

n(j)minus1 biji

y2 dσn(j)minus1

Snminus1(1)

n(j)sum

n(j)minus1(x)

n(j)sum

(n(j)minus 1)

Snminus1(1)

limjrarrinfin

(n(j)minus 1)

Snminus1(1)

We therefore have

limjrarrinfin

W2(σn(j)minus1

δo)2 = lim

jrarrinfin

n(j)sum

b2ij gt 0

radicn(j)minus1 bij

thatlimjrarrinfin

dP(ǫn(j)

radicn(j)minus1 bij

γn(j)

b2ij) = 0

We also have

limjrarrinfin

W2(γn(j)

b2ij δo)

2 = limjrarrinfin

jrarrinfin

n(j)sum

b2ij gt 0

limjrarrinfin

lim infjrarrinfin

n(j)sum

a2ij gt 0(ii)

n(j)sum

and bij =aij

maxAj 1

so that En(j)

lim infjrarrinfin

n(j)sum

b2ij gt 0 and

n(j)sum

b2ij le 1

fj En(j)

radicn(j)minus1 bij

Since en(j)

we see that en(j)

(minusBj) = en(j)

is box

is not an

(43) limjrarrinfin

n(j)sum

a2ij gt

infinsum

n(j)sum

a2ij minusinfinsum

Setting

aijk =

akj if i le k

aij if i ge k + 1

we have

limjrarrinfin

n(j)sum

a2ijk +

n(j)sum

a2ij minusksum

a2i gt ε0

n(j(k))sum

bullsumn(j(k))

References

1 Introduction

2 Preliminaries





25 Pyramid

26 Dissipation


28 Gaussian space



References

X to Y

X and Y by

25 Pyramid

alent to each other

trating to P

γna2i =

notimes

γ1a2i

where a2i are

We call PΓinfina2i

γinfina2i =

infinotimes

γ1a2i

(21)infinsum

a2i lt +infin

nminus1 of

αi σnminus1αi)

dominated by Enβi

converges weakly

in Subsection 28

lim infjrarrinfin

ObsDiam(En(j)

minusκ)

ObsDiam(En(j)

minusκ)

ge limjrarrinfin

1minus κ)

R = (nminus 1)12

Inminus1

tnminus1eminust2

int infin

tmeminust2

Elowastγ

ϕS(x) =

radicnminus 1

rx x isin Rn o

radicnminus1ai

radicn

with N le n

dϕEx le

radic1 + CNε

θand dϕS

x leradic1 + CNε

dϕEx(v)2 =

)partr

partxj〈x v〉+ R

〈x v〉2nsum

x2ja4j

r2part

)〈x v〉

vjxja2j

x2 = 1 +Nminus1sum

a2jminus 1

x2 = 1 +O(Nε2)

and so

x = 1 +O(Nε2)xar

= 1 +O(Nε2)

We also have

x2nsum

x2ja4j

= 1 +Nminus1sum

a4jminus 1

x2 = 1 + O(Nε2)

vjxja2j

vjxjx +

Nminus1sum

a2jminus 1

)vjxjx =

〈x v〉x +O(Nε)

) we have

dϕEx(v)2 = t2g2(1 +O(Nε2)) +

r(1 +O(Nε2))(31)

rO(Nε) +

2 dt we have

partr=

minus 1n

nand r ge θ

radicn yields

0 le partR

partrle

radicn

partrminus R

∣∣∣∣ le1

dϕEx(v)2 = t2g2 +

r2+O(θminus2Nε)

= t2(partR

+ (1minus t2)R2

r2+O(θminus2Nε)

limnrarrinfin

enradicnminus1ai(D

nNε) = 1(1)

limnrarrinfin

γna2i (Dn

nNε) = σnminus1

radicnminus1 ai

(DnNε) =

γna2i (Dn

ϕE if (En

radicnminus1ai e

nradicnminus1ai)

radicnminus1 ai σ

NεcapFn

|DnNε

capFnθ) and so

radicnminus1ai|Dn

Nε en

|DnNε

capFnθ γna2i

) rarr 0

PΓinfina2i

as nrarr infin

radicn(j)minus1 ai

Since En(j)

radicn(j)minus1 ai

radicnminus1 ai for

dropping

i dominates Γka2i

and so PΓinfina2i

supPΓinfin

a2i We thus obtain

which is a contra-

converges

The ellipsoid En(j)

converges

bεi =

ai if i le I(ε)

n(j)aiji ≺ E

n(j)(1+2ε)bεi = (1 + 2ε)E

PEn(j)

bεi =

2ε if i ge I(ε)

En(j)aiji ≺ E

in PΓinfinb2εi

I(ε)minus1

(1+ε)2a2i

and ΓkminusI(ε)+1

ρ(PΓkb2εiPΓ

I(ε)minus1

kb2εi

ΓI(ε)minus1

(1+ε)2a2i ) le τ(ε)

ρ(PΓinfinb2εi

PΓI(ε)minus1

(1+ε)2a2i) le τ(ε)

b2ε(l)i

I(ε(l))minus1

infin as

as l rarr infin

PΓinfina2i

does not converge

PEn(j)

i as j rarr infin

asymptotically

a2 is not

i contains PΓinfin

i is not

a2ij we have

ObsDiam(Γn(j)

a2ijconverges

a2i as j rarr

converges

so that

and so PΓn(j)

a2ijto PΓinfin

converges

such that

supaiisinA

infinsum

a2i lt +infin

Then we have

limnrarrinfin

supaiisinA

) = 0(1)

lim supnrarrinfin

supaiisinA

aiisinA

infinsum

a2i(2)

radicnminus1

radicnminus1 ])

radicnminus 1 )

We remark that

radicnminus 1 ]))

limnrarrinfin

wθn = 1

It then holds that

nθ ) le dTV(ǫ

nradicnminus1 ai ǫ

dP(γna2i

ψ(x) =R

radicnminus 1 ])

W2(ǫnθ γ

2 leint

rminus 1

x2 dγnθ (x)

radicnminus 1 ]))

x2 dγna2i (x)

and hence

lim supnrarrinfin

supaiisinA

aiisinA

infinsum

This proves (1)

radicnminus1ai

)2 leint

Snminus1(1)

a2ix2i dσ

nminus1(x) middot 1

Inminus1

int infin

Inminus1

int infin

22 dr le 1

Inminus1

simradic2eminus12

a2ix2i dσ

Snminus1kε

a2ix2i dσ

a2i +nsum

which imply

supaiisinA

Snminus1(1)

a2ix2i dσ

nminus1(x)

aiisinA

infinsum

a2i + supaiisinA

infinsum

limjrarrinfin

n(j)sum

(aij minus ai)2 = 0

Then we have

(1) ǫn(j)

(2) σn(j)minus1

j rarr infin

In particular En(j)

as j rarr infin

lim supjrarrinfin

W2(σn(j)minus1

γn(j)

a2iji)

jrarrinfin

infinsum

a2ij =radic2eminus1

infinsum

W2(γn(j)

a2iji γinfina2i

)2 =infinsum

radicn(j)minus1 bij

It holds that

W2(σn(j)minus1

δo)2 =

Sn(j)minus1

n(j)minus1 biji

y2 dσn(j)minus1

Snminus1(1)

n(j)sum

n(j)minus1(x)

n(j)sum

(n(j)minus 1)

Snminus1(1)

limjrarrinfin

(n(j)minus 1)

Snminus1(1)

We therefore have

limjrarrinfin

W2(σn(j)minus1

δo)2 = lim

jrarrinfin

n(j)sum

b2ij gt 0

radicn(j)minus1 bij

thatlimjrarrinfin

dP(ǫn(j)

radicn(j)minus1 bij

γn(j)

b2ij) = 0

We also have

limjrarrinfin

W2(γn(j)

b2ij δo)

2 = limjrarrinfin

jrarrinfin

n(j)sum

b2ij gt 0

limjrarrinfin

lim infjrarrinfin

n(j)sum

a2ij gt 0(ii)

n(j)sum

and bij =aij

maxAj 1

so that En(j)

lim infjrarrinfin

n(j)sum

b2ij gt 0 and

n(j)sum

b2ij le 1

fj En(j)

radicn(j)minus1 bij

Since en(j)

we see that en(j)

(minusBj) = en(j)

is box

is not an

(43) limjrarrinfin

n(j)sum

a2ij gt

infinsum

n(j)sum

a2ij minusinfinsum

Setting

aijk =

akj if i le k

aij if i ge k + 1

we have

limjrarrinfin

n(j)sum

a2ijk +

n(j)sum

a2ij minusksum

a2i gt ε0

n(j(k))sum

bullsumn(j(k))

References

1 Introduction

2 Preliminaries





25 Pyramid

26 Dissipation


28 Gaussian space



References

25 Pyramid

alent to each other

trating to P

γna2i =

notimes

γ1a2i

where a2i are

We call PΓinfina2i

γinfina2i =

infinotimes

γ1a2i

(21)infinsum

a2i lt +infin

nminus1 of

αi σnminus1αi)

dominated by Enβi

converges weakly

in Subsection 28

lim infjrarrinfin

ObsDiam(En(j)

minusκ)

ObsDiam(En(j)

minusκ)

ge limjrarrinfin

1minus κ)

R = (nminus 1)12

Inminus1

tnminus1eminust2

int infin

tmeminust2

Elowastγ

ϕS(x) =

radicnminus 1

rx x isin Rn o

radicnminus1ai

radicn

with N le n

dϕEx le

radic1 + CNε

θand dϕS

x leradic1 + CNε

dϕEx(v)2 =

)partr

partxj〈x v〉+ R

〈x v〉2nsum

x2ja4j

r2part

)〈x v〉

vjxja2j

x2 = 1 +Nminus1sum

a2jminus 1

x2 = 1 +O(Nε2)

and so

x = 1 +O(Nε2)xar

= 1 +O(Nε2)

We also have

x2nsum

x2ja4j

= 1 +Nminus1sum

a4jminus 1

x2 = 1 + O(Nε2)

vjxja2j

vjxjx +

Nminus1sum

a2jminus 1

)vjxjx =

〈x v〉x +O(Nε)

) we have

dϕEx(v)2 = t2g2(1 +O(Nε2)) +

r(1 +O(Nε2))(31)

rO(Nε) +

2 dt we have

partr=

minus 1n

nand r ge θ

radicn yields

0 le partR

partrle

radicn

partrminus R

∣∣∣∣ le1

dϕEx(v)2 = t2g2 +

r2+O(θminus2Nε)

= t2(partR

+ (1minus t2)R2

r2+O(θminus2Nε)

limnrarrinfin

enradicnminus1ai(D

nNε) = 1(1)

limnrarrinfin

γna2i (Dn

nNε) = σnminus1

radicnminus1 ai

(DnNε) =

γna2i (Dn

ϕE if (En

radicnminus1ai e

nradicnminus1ai)

radicnminus1 ai σ

NεcapFn

|DnNε

capFnθ) and so

radicnminus1ai|Dn

Nε en

|DnNε

capFnθ γna2i

) rarr 0

PΓinfina2i

as nrarr infin

radicn(j)minus1 ai

Since En(j)

radicn(j)minus1 ai

radicnminus1 ai for

dropping

i dominates Γka2i

and so PΓinfina2i

supPΓinfin

a2i We thus obtain

which is a contra-

converges

The ellipsoid En(j)

converges

bεi =

ai if i le I(ε)

n(j)aiji ≺ E

n(j)(1+2ε)bεi = (1 + 2ε)E

PEn(j)

bεi =

2ε if i ge I(ε)

En(j)aiji ≺ E

in PΓinfinb2εi

I(ε)minus1

(1+ε)2a2i

and ΓkminusI(ε)+1

ρ(PΓkb2εiPΓ

I(ε)minus1

kb2εi

ΓI(ε)minus1

(1+ε)2a2i ) le τ(ε)

ρ(PΓinfinb2εi

PΓI(ε)minus1

(1+ε)2a2i) le τ(ε)

b2ε(l)i

I(ε(l))minus1

infin as

as l rarr infin

PΓinfina2i

does not converge

PEn(j)

i as j rarr infin

asymptotically

a2 is not

i contains PΓinfin

i is not

a2ij we have

ObsDiam(Γn(j)

a2ijconverges

a2i as j rarr

converges

so that

and so PΓn(j)

a2ijto PΓinfin

converges

such that

supaiisinA

infinsum

a2i lt +infin

Then we have

limnrarrinfin

supaiisinA

) = 0(1)

lim supnrarrinfin

supaiisinA

aiisinA

infinsum

a2i(2)

radicnminus1

radicnminus1 ])

radicnminus 1 )

We remark that

radicnminus 1 ]))

limnrarrinfin

wθn = 1

It then holds that

nθ ) le dTV(ǫ

nradicnminus1 ai ǫ

dP(γna2i

ψ(x) =R

radicnminus 1 ])

W2(ǫnθ γ

2 leint

rminus 1

x2 dγnθ (x)

radicnminus 1 ]))

x2 dγna2i (x)

and hence

lim supnrarrinfin

supaiisinA

aiisinA

infinsum

This proves (1)

radicnminus1ai

)2 leint

Snminus1(1)

a2ix2i dσ

nminus1(x) middot 1

Inminus1

int infin

Inminus1

int infin

22 dr le 1

Inminus1

simradic2eminus12

a2ix2i dσ

Snminus1kε

a2ix2i dσ

a2i +nsum

which imply

supaiisinA

Snminus1(1)

a2ix2i dσ

nminus1(x)

aiisinA

infinsum

a2i + supaiisinA

infinsum

limjrarrinfin

n(j)sum

(aij minus ai)2 = 0

Then we have

(1) ǫn(j)

(2) σn(j)minus1

j rarr infin

In particular En(j)

as j rarr infin

lim supjrarrinfin

W2(σn(j)minus1

γn(j)

a2iji)

jrarrinfin

infinsum

a2ij =radic2eminus1

infinsum

W2(γn(j)

a2iji γinfina2i

)2 =infinsum

radicn(j)minus1 bij

It holds that

W2(σn(j)minus1

δo)2 =

Sn(j)minus1

n(j)minus1 biji

y2 dσn(j)minus1

Snminus1(1)

n(j)sum

n(j)minus1(x)

n(j)sum

(n(j)minus 1)

Snminus1(1)

limjrarrinfin

(n(j)minus 1)

Snminus1(1)

We therefore have

limjrarrinfin

W2(σn(j)minus1

δo)2 = lim

jrarrinfin

n(j)sum

b2ij gt 0

radicn(j)minus1 bij

thatlimjrarrinfin

dP(ǫn(j)

radicn(j)minus1 bij

γn(j)

b2ij) = 0

We also have

limjrarrinfin

W2(γn(j)

b2ij δo)

2 = limjrarrinfin

jrarrinfin

n(j)sum

b2ij gt 0

limjrarrinfin

lim infjrarrinfin

n(j)sum

a2ij gt 0(ii)

n(j)sum

and bij =aij

maxAj 1

so that En(j)

lim infjrarrinfin

n(j)sum

b2ij gt 0 and

n(j)sum

b2ij le 1

fj En(j)

radicn(j)minus1 bij

Since en(j)

we see that en(j)

(minusBj) = en(j)

is box

is not an

(43) limjrarrinfin

n(j)sum

a2ij gt

infinsum

n(j)sum

a2ij minusinfinsum

Setting

aijk =

akj if i le k

aij if i ge k + 1

we have

limjrarrinfin

n(j)sum

a2ijk +

n(j)sum

a2ij minusksum

a2i gt ε0

n(j(k))sum

bullsumn(j(k))

References

1 Introduction

2 Preliminaries





25 Pyramid

26 Dissipation


28 Gaussian space



References

alent to each other

trating to P

γna2i =

notimes

γ1a2i

where a2i are

We call PΓinfina2i

γinfina2i =

infinotimes

γ1a2i

(21)infinsum

a2i lt +infin

nminus1 of

αi σnminus1αi)

dominated by Enβi

converges weakly

in Subsection 28

lim infjrarrinfin

ObsDiam(En(j)

minusκ)

ObsDiam(En(j)

minusκ)

ge limjrarrinfin

1minus κ)

R = (nminus 1)12

Inminus1

tnminus1eminust2

int infin

tmeminust2

Elowastγ

ϕS(x) =

radicnminus 1

rx x isin Rn o

radicnminus1ai

radicn

with N le n

dϕEx le

radic1 + CNε

θand dϕS

x leradic1 + CNε

dϕEx(v)2 =

)partr

partxj〈x v〉+ R

〈x v〉2nsum

x2ja4j

r2part

)〈x v〉

vjxja2j

x2 = 1 +Nminus1sum

a2jminus 1

x2 = 1 +O(Nε2)

and so

x = 1 +O(Nε2)xar

= 1 +O(Nε2)

We also have

x2nsum

x2ja4j

= 1 +Nminus1sum

a4jminus 1

x2 = 1 + O(Nε2)

vjxja2j

vjxjx +

Nminus1sum

a2jminus 1

)vjxjx =

〈x v〉x +O(Nε)

) we have

dϕEx(v)2 = t2g2(1 +O(Nε2)) +

r(1 +O(Nε2))(31)

rO(Nε) +

2 dt we have

partr=

minus 1n

nand r ge θ

radicn yields

0 le partR

partrle

radicn

partrminus R

∣∣∣∣ le1

dϕEx(v)2 = t2g2 +

r2+O(θminus2Nε)

= t2(partR

+ (1minus t2)R2

r2+O(θminus2Nε)

limnrarrinfin

enradicnminus1ai(D

nNε) = 1(1)

limnrarrinfin

γna2i (Dn

nNε) = σnminus1

radicnminus1 ai

(DnNε) =

γna2i (Dn

ϕE if (En

radicnminus1ai e

nradicnminus1ai)

radicnminus1 ai σ

NεcapFn

|DnNε

capFnθ) and so

radicnminus1ai|Dn

Nε en

|DnNε

capFnθ γna2i

) rarr 0

PΓinfina2i

as nrarr infin

radicn(j)minus1 ai

Since En(j)

radicn(j)minus1 ai

radicnminus1 ai for

dropping

i dominates Γka2i

and so PΓinfina2i

supPΓinfin

a2i We thus obtain

which is a contra-

converges

The ellipsoid En(j)

converges

bεi =

ai if i le I(ε)

n(j)aiji ≺ E

n(j)(1+2ε)bεi = (1 + 2ε)E

PEn(j)

bεi =

2ε if i ge I(ε)

En(j)aiji ≺ E

in PΓinfinb2εi

I(ε)minus1

(1+ε)2a2i

and ΓkminusI(ε)+1

ρ(PΓkb2εiPΓ

I(ε)minus1

kb2εi

ΓI(ε)minus1

(1+ε)2a2i ) le τ(ε)

ρ(PΓinfinb2εi

PΓI(ε)minus1

(1+ε)2a2i) le τ(ε)

b2ε(l)i

I(ε(l))minus1

infin as

as l rarr infin

PΓinfina2i

does not converge

PEn(j)

i as j rarr infin

asymptotically

a2 is not

i contains PΓinfin

i is not

a2ij we have

ObsDiam(Γn(j)

a2ijconverges

a2i as j rarr

converges

so that

and so PΓn(j)

a2ijto PΓinfin

converges

such that

supaiisinA

infinsum

a2i lt +infin

Then we have

limnrarrinfin

supaiisinA

) = 0(1)

lim supnrarrinfin

supaiisinA

aiisinA

infinsum

a2i(2)

radicnminus1

radicnminus1 ])

radicnminus 1 )

We remark that

radicnminus 1 ]))

limnrarrinfin

wθn = 1

It then holds that

nθ ) le dTV(ǫ

nradicnminus1 ai ǫ

dP(γna2i

ψ(x) =R

radicnminus 1 ])

W2(ǫnθ γ

2 leint

rminus 1

x2 dγnθ (x)

radicnminus 1 ]))

x2 dγna2i (x)

and hence

lim supnrarrinfin

supaiisinA

aiisinA

infinsum

This proves (1)

radicnminus1ai

)2 leint

Snminus1(1)

a2ix2i dσ

nminus1(x) middot 1

Inminus1

int infin

Inminus1

int infin

22 dr le 1

Inminus1

simradic2eminus12

a2ix2i dσ

Snminus1kε

a2ix2i dσ

a2i +nsum

which imply

supaiisinA

Snminus1(1)

a2ix2i dσ

nminus1(x)

aiisinA

infinsum

a2i + supaiisinA

infinsum

limjrarrinfin

n(j)sum

(aij minus ai)2 = 0

Then we have

(1) ǫn(j)

(2) σn(j)minus1

j rarr infin

In particular En(j)

as j rarr infin

lim supjrarrinfin

W2(σn(j)minus1

γn(j)

a2iji)

jrarrinfin

infinsum

a2ij =radic2eminus1

infinsum

W2(γn(j)

a2iji γinfina2i

)2 =infinsum

radicn(j)minus1 bij

It holds that

W2(σn(j)minus1

δo)2 =

Sn(j)minus1

n(j)minus1 biji

y2 dσn(j)minus1

Snminus1(1)

n(j)sum

n(j)minus1(x)

n(j)sum

(n(j)minus 1)

Snminus1(1)

limjrarrinfin

(n(j)minus 1)

Snminus1(1)

We therefore have

limjrarrinfin

W2(σn(j)minus1

δo)2 = lim

jrarrinfin

n(j)sum

b2ij gt 0

radicn(j)minus1 bij

thatlimjrarrinfin

dP(ǫn(j)

radicn(j)minus1 bij

γn(j)

b2ij) = 0

We also have

limjrarrinfin

W2(γn(j)

b2ij δo)

2 = limjrarrinfin

jrarrinfin

n(j)sum

b2ij gt 0

limjrarrinfin

lim infjrarrinfin

n(j)sum

a2ij gt 0(ii)

n(j)sum

and bij =aij

maxAj 1

so that En(j)

lim infjrarrinfin

n(j)sum

b2ij gt 0 and

n(j)sum

b2ij le 1

fj En(j)

radicn(j)minus1 bij

Since en(j)

we see that en(j)

(minusBj) = en(j)

is box

is not an

(43) limjrarrinfin

n(j)sum

a2ij gt

infinsum

n(j)sum

a2ij minusinfinsum

Setting

aijk =

akj if i le k

aij if i ge k + 1

we have

limjrarrinfin

n(j)sum

a2ijk +

n(j)sum

a2ij minusksum

a2i gt ε0

n(j(k))sum

bullsumn(j(k))

References

1 Introduction

2 Preliminaries





25 Pyramid

26 Dissipation


28 Gaussian space



References

We call PΓinfina2i

γinfina2i =

infinotimes

γ1a2i

(21)infinsum

a2i lt +infin

nminus1 of

αi σnminus1αi)

dominated by Enβi

converges weakly

in Subsection 28

lim infjrarrinfin

ObsDiam(En(j)

minusκ)

ObsDiam(En(j)

minusκ)

ge limjrarrinfin

1minus κ)

R = (nminus 1)12

Inminus1

tnminus1eminust2

int infin

tmeminust2

Elowastγ

ϕS(x) =

radicnminus 1

rx x isin Rn o

radicnminus1ai

radicn

with N le n

dϕEx le

radic1 + CNε

θand dϕS

x leradic1 + CNε

dϕEx(v)2 =

)partr

partxj〈x v〉+ R

〈x v〉2nsum

x2ja4j

r2part

)〈x v〉

vjxja2j

x2 = 1 +Nminus1sum

a2jminus 1

x2 = 1 +O(Nε2)

and so

x = 1 +O(Nε2)xar

= 1 +O(Nε2)

We also have

x2nsum

x2ja4j

= 1 +Nminus1sum

a4jminus 1

x2 = 1 + O(Nε2)

vjxja2j

vjxjx +

Nminus1sum

a2jminus 1

)vjxjx =

〈x v〉x +O(Nε)

) we have

dϕEx(v)2 = t2g2(1 +O(Nε2)) +

r(1 +O(Nε2))(31)

rO(Nε) +

2 dt we have

partr=

minus 1n

nand r ge θ

radicn yields

0 le partR

partrle

radicn

partrminus R

∣∣∣∣ le1

dϕEx(v)2 = t2g2 +

r2+O(θminus2Nε)

= t2(partR

+ (1minus t2)R2

r2+O(θminus2Nε)

limnrarrinfin

enradicnminus1ai(D

nNε) = 1(1)

limnrarrinfin

γna2i (Dn

nNε) = σnminus1

radicnminus1 ai

(DnNε) =

γna2i (Dn

ϕE if (En

radicnminus1ai e

nradicnminus1ai)

radicnminus1 ai σ

NεcapFn

|DnNε

capFnθ) and so

radicnminus1ai|Dn

Nε en

|DnNε

capFnθ γna2i

) rarr 0

PΓinfina2i

as nrarr infin

radicn(j)minus1 ai

Since En(j)

radicn(j)minus1 ai

radicnminus1 ai for

dropping

i dominates Γka2i

and so PΓinfina2i

supPΓinfin

a2i We thus obtain

which is a contra-

converges

The ellipsoid En(j)

converges

bεi =

ai if i le I(ε)

n(j)aiji ≺ E

n(j)(1+2ε)bεi = (1 + 2ε)E

PEn(j)

bεi =

2ε if i ge I(ε)

En(j)aiji ≺ E

in PΓinfinb2εi

I(ε)minus1

(1+ε)2a2i

and ΓkminusI(ε)+1

ρ(PΓkb2εiPΓ

I(ε)minus1

kb2εi

ΓI(ε)minus1

(1+ε)2a2i ) le τ(ε)

ρ(PΓinfinb2εi

PΓI(ε)minus1

(1+ε)2a2i) le τ(ε)

b2ε(l)i

I(ε(l))minus1

infin as

as l rarr infin

PΓinfina2i

does not converge

PEn(j)

i as j rarr infin

asymptotically

a2 is not

i contains PΓinfin

i is not

a2ij we have

ObsDiam(Γn(j)

a2ijconverges

a2i as j rarr

converges

so that

and so PΓn(j)

a2ijto PΓinfin

converges

such that

supaiisinA

infinsum

a2i lt +infin

Then we have

limnrarrinfin

supaiisinA

) = 0(1)

lim supnrarrinfin

supaiisinA

aiisinA

infinsum

a2i(2)

radicnminus1

radicnminus1 ])

radicnminus 1 )

We remark that

radicnminus 1 ]))

limnrarrinfin

wθn = 1

It then holds that

nθ ) le dTV(ǫ

nradicnminus1 ai ǫ

dP(γna2i

ψ(x) =R

radicnminus 1 ])

W2(ǫnθ γ

2 leint

rminus 1

x2 dγnθ (x)

radicnminus 1 ]))

x2 dγna2i (x)

and hence

lim supnrarrinfin

supaiisinA

aiisinA

infinsum

This proves (1)

radicnminus1ai

)2 leint

Snminus1(1)

a2ix2i dσ

nminus1(x) middot 1

Inminus1

int infin

Inminus1

int infin

22 dr le 1

Inminus1

simradic2eminus12

a2ix2i dσ

Snminus1kε

a2ix2i dσ

a2i +nsum

which imply

supaiisinA

Snminus1(1)

a2ix2i dσ

nminus1(x)

aiisinA

infinsum

a2i + supaiisinA

infinsum

limjrarrinfin

n(j)sum

(aij minus ai)2 = 0

Then we have

(1) ǫn(j)

(2) σn(j)minus1

j rarr infin

In particular En(j)

as j rarr infin

lim supjrarrinfin

W2(σn(j)minus1

γn(j)

a2iji)

jrarrinfin

infinsum

a2ij =radic2eminus1

infinsum

W2(γn(j)

a2iji γinfina2i

)2 =infinsum

radicn(j)minus1 bij

It holds that

W2(σn(j)minus1

δo)2 =

Sn(j)minus1

n(j)minus1 biji

y2 dσn(j)minus1

Snminus1(1)

n(j)sum

n(j)minus1(x)

n(j)sum

(n(j)minus 1)

Snminus1(1)

limjrarrinfin

(n(j)minus 1)

Snminus1(1)

We therefore have

limjrarrinfin

W2(σn(j)minus1

δo)2 = lim

jrarrinfin

n(j)sum

b2ij gt 0

radicn(j)minus1 bij

thatlimjrarrinfin

dP(ǫn(j)

radicn(j)minus1 bij

γn(j)

b2ij) = 0

We also have

limjrarrinfin

W2(γn(j)

b2ij δo)

2 = limjrarrinfin

jrarrinfin

n(j)sum

b2ij gt 0

limjrarrinfin

lim infjrarrinfin

n(j)sum

a2ij gt 0(ii)

n(j)sum

and bij =aij

maxAj 1

so that En(j)

lim infjrarrinfin

n(j)sum

b2ij gt 0 and

n(j)sum

b2ij le 1

fj En(j)

radicn(j)minus1 bij

Since en(j)

we see that en(j)

(minusBj) = en(j)

is box

is not an

(43) limjrarrinfin

n(j)sum

a2ij gt

infinsum

n(j)sum

a2ij minusinfinsum

Setting

aijk =

akj if i le k

aij if i ge k + 1

we have

limjrarrinfin

n(j)sum

a2ijk +

n(j)sum

a2ij minusksum

a2i gt ε0

n(j(k))sum

bullsumn(j(k))

References

1 Introduction

2 Preliminaries





25 Pyramid

26 Dissipation


28 Gaussian space



References

dominated by Enβi

converges weakly

in Subsection 28

lim infjrarrinfin

ObsDiam(En(j)

minusκ)

ObsDiam(En(j)

minusκ)

ge limjrarrinfin

1minus κ)

R = (nminus 1)12

Inminus1

tnminus1eminust2

int infin

tmeminust2

Elowastγ

ϕS(x) =

radicnminus 1

rx x isin Rn o

radicnminus1ai

radicn

with N le n

dϕEx le

radic1 + CNε

θand dϕS

x leradic1 + CNε

dϕEx(v)2 =

)partr

partxj〈x v〉+ R

〈x v〉2nsum

x2ja4j

r2part

)〈x v〉

vjxja2j

x2 = 1 +Nminus1sum

a2jminus 1

x2 = 1 +O(Nε2)

and so

x = 1 +O(Nε2)xar

= 1 +O(Nε2)

We also have

x2nsum

x2ja4j

= 1 +Nminus1sum

a4jminus 1

x2 = 1 + O(Nε2)

vjxja2j

vjxjx +

Nminus1sum

a2jminus 1

)vjxjx =

〈x v〉x +O(Nε)

) we have

dϕEx(v)2 = t2g2(1 +O(Nε2)) +

r(1 +O(Nε2))(31)

rO(Nε) +

2 dt we have

partr=

minus 1n

nand r ge θ

radicn yields

0 le partR

partrle

radicn

partrminus R

∣∣∣∣ le1

dϕEx(v)2 = t2g2 +

r2+O(θminus2Nε)

= t2(partR

+ (1minus t2)R2

r2+O(θminus2Nε)

limnrarrinfin

enradicnminus1ai(D

nNε) = 1(1)

limnrarrinfin

γna2i (Dn

nNε) = σnminus1

radicnminus1 ai

(DnNε) =

γna2i (Dn

ϕE if (En

radicnminus1ai e

nradicnminus1ai)

radicnminus1 ai σ

NεcapFn

|DnNε

capFnθ) and so

radicnminus1ai|Dn

Nε en

|DnNε

capFnθ γna2i

) rarr 0

PΓinfina2i

as nrarr infin

radicn(j)minus1 ai

Since En(j)

radicn(j)minus1 ai

radicnminus1 ai for

dropping

i dominates Γka2i

and so PΓinfina2i

supPΓinfin

a2i We thus obtain

which is a contra-

converges

The ellipsoid En(j)

converges

bεi =

ai if i le I(ε)

n(j)aiji ≺ E

n(j)(1+2ε)bεi = (1 + 2ε)E

PEn(j)

bεi =

2ε if i ge I(ε)

En(j)aiji ≺ E

in PΓinfinb2εi

I(ε)minus1

(1+ε)2a2i

and ΓkminusI(ε)+1

ρ(PΓkb2εiPΓ

I(ε)minus1

kb2εi

ΓI(ε)minus1

(1+ε)2a2i ) le τ(ε)

ρ(PΓinfinb2εi

PΓI(ε)minus1

(1+ε)2a2i) le τ(ε)

b2ε(l)i

I(ε(l))minus1

infin as

as l rarr infin

PΓinfina2i

does not converge

PEn(j)

i as j rarr infin

asymptotically

a2 is not

i contains PΓinfin

i is not

a2ij we have

ObsDiam(Γn(j)

a2ijconverges

a2i as j rarr

converges

so that

and so PΓn(j)

a2ijto PΓinfin

converges

such that

supaiisinA

infinsum

a2i lt +infin

Then we have

limnrarrinfin

supaiisinA

) = 0(1)

lim supnrarrinfin

supaiisinA

aiisinA

infinsum

a2i(2)

radicnminus1

radicnminus1 ])

radicnminus 1 )

We remark that

radicnminus 1 ]))

limnrarrinfin

wθn = 1

It then holds that

nθ ) le dTV(ǫ

nradicnminus1 ai ǫ

dP(γna2i

ψ(x) =R

radicnminus 1 ])

W2(ǫnθ γ

2 leint

rminus 1

x2 dγnθ (x)

radicnminus 1 ]))

x2 dγna2i (x)

and hence

lim supnrarrinfin

supaiisinA

aiisinA

infinsum

This proves (1)

radicnminus1ai

)2 leint

Snminus1(1)

a2ix2i dσ

nminus1(x) middot 1

Inminus1

int infin

Inminus1

int infin

22 dr le 1

Inminus1

simradic2eminus12

a2ix2i dσ

Snminus1kε

a2ix2i dσ

a2i +nsum

which imply

supaiisinA

Snminus1(1)

a2ix2i dσ

nminus1(x)

aiisinA

infinsum

a2i + supaiisinA

infinsum

limjrarrinfin

n(j)sum

(aij minus ai)2 = 0

Then we have

(1) ǫn(j)

(2) σn(j)minus1

j rarr infin

In particular En(j)

as j rarr infin

lim supjrarrinfin

W2(σn(j)minus1

γn(j)

a2iji)

jrarrinfin

infinsum

a2ij =radic2eminus1

infinsum

W2(γn(j)

a2iji γinfina2i

)2 =infinsum

radicn(j)minus1 bij

It holds that

W2(σn(j)minus1

δo)2 =

Sn(j)minus1

n(j)minus1 biji

y2 dσn(j)minus1

Snminus1(1)

n(j)sum

n(j)minus1(x)

n(j)sum

(n(j)minus 1)

Snminus1(1)

limjrarrinfin

(n(j)minus 1)

Snminus1(1)

We therefore have

limjrarrinfin

W2(σn(j)minus1

δo)2 = lim

jrarrinfin

n(j)sum

b2ij gt 0

radicn(j)minus1 bij

thatlimjrarrinfin

dP(ǫn(j)

radicn(j)minus1 bij

γn(j)

b2ij) = 0

We also have

limjrarrinfin

W2(γn(j)

b2ij δo)

2 = limjrarrinfin

jrarrinfin

n(j)sum

b2ij gt 0

limjrarrinfin

lim infjrarrinfin

n(j)sum

a2ij gt 0(ii)

n(j)sum

and bij =aij

maxAj 1

so that En(j)

lim infjrarrinfin

n(j)sum

b2ij gt 0 and

n(j)sum

b2ij le 1

fj En(j)

radicn(j)minus1 bij

Since en(j)

we see that en(j)

(minusBj) = en(j)

is box

is not an

(43) limjrarrinfin

n(j)sum

a2ij gt

infinsum

n(j)sum

a2ij minusinfinsum

Setting

aijk =

akj if i le k

aij if i ge k + 1

we have

limjrarrinfin

n(j)sum

a2ijk +

n(j)sum

a2ij minusksum

a2i gt ε0

n(j(k))sum

bullsumn(j(k))

References

1 Introduction

2 Preliminaries





25 Pyramid

26 Dissipation


28 Gaussian space



References

R = (nminus 1)12

Inminus1

tnminus1eminust2

int infin

tmeminust2

Elowastγ

ϕS(x) =

radicnminus 1

rx x isin Rn o

radicnminus1ai

radicn

with N le n

dϕEx le

radic1 + CNε

θand dϕS

x leradic1 + CNε

dϕEx(v)2 =

)partr

partxj〈x v〉+ R

〈x v〉2nsum

x2ja4j

r2part

)〈x v〉

vjxja2j

x2 = 1 +Nminus1sum

a2jminus 1

x2 = 1 +O(Nε2)

and so

x = 1 +O(Nε2)xar

= 1 +O(Nε2)

We also have

x2nsum

x2ja4j

= 1 +Nminus1sum

a4jminus 1

x2 = 1 + O(Nε2)

vjxja2j

vjxjx +

Nminus1sum

a2jminus 1

)vjxjx =

〈x v〉x +O(Nε)

) we have

dϕEx(v)2 = t2g2(1 +O(Nε2)) +

r(1 +O(Nε2))(31)

rO(Nε) +

2 dt we have

partr=

minus 1n

nand r ge θ

radicn yields

0 le partR

partrle

radicn

partrminus R

∣∣∣∣ le1

dϕEx(v)2 = t2g2 +

r2+O(θminus2Nε)

= t2(partR

+ (1minus t2)R2

r2+O(θminus2Nε)

limnrarrinfin

enradicnminus1ai(D

nNε) = 1(1)

limnrarrinfin

γna2i (Dn

nNε) = σnminus1

radicnminus1 ai

(DnNε) =

γna2i (Dn

ϕE if (En

radicnminus1ai e

nradicnminus1ai)

radicnminus1 ai σ

NεcapFn

|DnNε

capFnθ) and so

radicnminus1ai|Dn

Nε en

|DnNε

capFnθ γna2i

) rarr 0

PΓinfina2i

as nrarr infin

radicn(j)minus1 ai

Since En(j)

radicn(j)minus1 ai

radicnminus1 ai for

dropping

i dominates Γka2i

and so PΓinfina2i

supPΓinfin

a2i We thus obtain

which is a contra-

converges

The ellipsoid En(j)

converges

bεi =

ai if i le I(ε)

n(j)aiji ≺ E

n(j)(1+2ε)bεi = (1 + 2ε)E

PEn(j)

bεi =

2ε if i ge I(ε)

En(j)aiji ≺ E

in PΓinfinb2εi

I(ε)minus1

(1+ε)2a2i

and ΓkminusI(ε)+1

ρ(PΓkb2εiPΓ

I(ε)minus1

kb2εi

ΓI(ε)minus1

(1+ε)2a2i ) le τ(ε)

ρ(PΓinfinb2εi

PΓI(ε)minus1

(1+ε)2a2i) le τ(ε)

b2ε(l)i

I(ε(l))minus1

infin as

as l rarr infin

PΓinfina2i

does not converge

PEn(j)

i as j rarr infin

asymptotically

a2 is not

i contains PΓinfin

i is not

a2ij we have

ObsDiam(Γn(j)

a2ijconverges

a2i as j rarr

converges

so that

and so PΓn(j)

a2ijto PΓinfin

converges

such that

supaiisinA

infinsum

a2i lt +infin

Then we have

limnrarrinfin

supaiisinA

) = 0(1)

lim supnrarrinfin

supaiisinA

aiisinA

infinsum

a2i(2)

radicnminus1

radicnminus1 ])

radicnminus 1 )

We remark that

radicnminus 1 ]))

limnrarrinfin

wθn = 1

It then holds that

nθ ) le dTV(ǫ

nradicnminus1 ai ǫ

dP(γna2i

ψ(x) =R

radicnminus 1 ])

W2(ǫnθ γ

2 leint

rminus 1

x2 dγnθ (x)

radicnminus 1 ]))

x2 dγna2i (x)

and hence

lim supnrarrinfin

supaiisinA

aiisinA

infinsum

This proves (1)

radicnminus1ai

)2 leint

Snminus1(1)

a2ix2i dσ

nminus1(x) middot 1

Inminus1

int infin

Inminus1

int infin

22 dr le 1

Inminus1

simradic2eminus12

a2ix2i dσ

Snminus1kε

a2ix2i dσ

a2i +nsum

which imply

supaiisinA

Snminus1(1)

a2ix2i dσ

nminus1(x)

aiisinA

infinsum

a2i + supaiisinA

infinsum

limjrarrinfin

n(j)sum

(aij minus ai)2 = 0

Then we have

(1) ǫn(j)

(2) σn(j)minus1

j rarr infin

In particular En(j)

as j rarr infin

lim supjrarrinfin

W2(σn(j)minus1

γn(j)

a2iji)

jrarrinfin

infinsum

a2ij =radic2eminus1

infinsum

W2(γn(j)

a2iji γinfina2i

)2 =infinsum

radicn(j)minus1 bij

It holds that

W2(σn(j)minus1

δo)2 =

Sn(j)minus1

n(j)minus1 biji

y2 dσn(j)minus1

Snminus1(1)

n(j)sum

n(j)minus1(x)

n(j)sum

(n(j)minus 1)

Snminus1(1)

limjrarrinfin

(n(j)minus 1)

Snminus1(1)

We therefore have

limjrarrinfin

W2(σn(j)minus1

δo)2 = lim

jrarrinfin

n(j)sum

b2ij gt 0

radicn(j)minus1 bij

thatlimjrarrinfin

dP(ǫn(j)

radicn(j)minus1 bij

γn(j)

b2ij) = 0

We also have

limjrarrinfin

W2(γn(j)

b2ij δo)

2 = limjrarrinfin

jrarrinfin

n(j)sum

b2ij gt 0

limjrarrinfin

lim infjrarrinfin

n(j)sum

a2ij gt 0(ii)

n(j)sum

and bij =aij

maxAj 1

so that En(j)

lim infjrarrinfin

n(j)sum

b2ij gt 0 and

n(j)sum

b2ij le 1

fj En(j)

radicn(j)minus1 bij

Since en(j)

we see that en(j)

(minusBj) = en(j)

is box

is not an

(43) limjrarrinfin

n(j)sum

a2ij gt

infinsum

n(j)sum

a2ij minusinfinsum

Setting

aijk =

akj if i le k

aij if i ge k + 1

we have

limjrarrinfin

n(j)sum

a2ijk +

n(j)sum

a2ij minusksum

a2i gt ε0

n(j(k))sum

bullsumn(j(k))

References

1 Introduction

2 Preliminaries





25 Pyramid

26 Dissipation


28 Gaussian space



References

dϕEx(v)2 =

)partr

partxj〈x v〉+ R

〈x v〉2nsum

x2ja4j

r2part

)〈x v〉

vjxja2j

x2 = 1 +Nminus1sum

a2jminus 1

x2 = 1 +O(Nε2)

and so

x = 1 +O(Nε2)xar

= 1 +O(Nε2)

We also have

x2nsum

x2ja4j

= 1 +Nminus1sum

a4jminus 1

x2 = 1 + O(Nε2)

vjxja2j

vjxjx +

Nminus1sum

a2jminus 1

)vjxjx =

〈x v〉x +O(Nε)

) we have

dϕEx(v)2 = t2g2(1 +O(Nε2)) +

r(1 +O(Nε2))(31)

rO(Nε) +

2 dt we have

partr=

minus 1n

nand r ge θ

radicn yields

0 le partR

partrle

radicn

partrminus R

∣∣∣∣ le1

dϕEx(v)2 = t2g2 +

r2+O(θminus2Nε)

= t2(partR

+ (1minus t2)R2

r2+O(θminus2Nε)

limnrarrinfin

enradicnminus1ai(D

nNε) = 1(1)

limnrarrinfin

γna2i (Dn

nNε) = σnminus1

radicnminus1 ai

(DnNε) =

γna2i (Dn

ϕE if (En

radicnminus1ai e

nradicnminus1ai)

radicnminus1 ai σ

NεcapFn

|DnNε

capFnθ) and so

radicnminus1ai|Dn

Nε en

|DnNε

capFnθ γna2i

) rarr 0

PΓinfina2i

as nrarr infin

radicn(j)minus1 ai

Since En(j)

radicn(j)minus1 ai

radicnminus1 ai for

dropping

i dominates Γka2i

and so PΓinfina2i

supPΓinfin

a2i We thus obtain

which is a contra-

converges

The ellipsoid En(j)

converges

bεi =

ai if i le I(ε)

n(j)aiji ≺ E

n(j)(1+2ε)bεi = (1 + 2ε)E

PEn(j)

bεi =

2ε if i ge I(ε)

En(j)aiji ≺ E

in PΓinfinb2εi

I(ε)minus1

(1+ε)2a2i

and ΓkminusI(ε)+1

ρ(PΓkb2εiPΓ

I(ε)minus1

kb2εi

ΓI(ε)minus1

(1+ε)2a2i ) le τ(ε)

ρ(PΓinfinb2εi

PΓI(ε)minus1

(1+ε)2a2i) le τ(ε)

b2ε(l)i

I(ε(l))minus1

infin as

as l rarr infin

PΓinfina2i

does not converge

PEn(j)

i as j rarr infin

asymptotically

a2 is not

i contains PΓinfin

i is not

a2ij we have

ObsDiam(Γn(j)

a2ijconverges

a2i as j rarr

converges

so that

and so PΓn(j)

a2ijto PΓinfin

converges

such that

supaiisinA

infinsum

a2i lt +infin

Then we have

limnrarrinfin

supaiisinA

) = 0(1)

lim supnrarrinfin

supaiisinA

aiisinA

infinsum

a2i(2)

radicnminus1

radicnminus1 ])

radicnminus 1 )

We remark that

radicnminus 1 ]))

limnrarrinfin

wθn = 1

It then holds that

nθ ) le dTV(ǫ

nradicnminus1 ai ǫ

dP(γna2i

ψ(x) =R

radicnminus 1 ])

W2(ǫnθ γ

2 leint

rminus 1

x2 dγnθ (x)

radicnminus 1 ]))

x2 dγna2i (x)

and hence

lim supnrarrinfin

supaiisinA

aiisinA

infinsum

This proves (1)

radicnminus1ai

)2 leint

Snminus1(1)

a2ix2i dσ

nminus1(x) middot 1

Inminus1

int infin

Inminus1

int infin

22 dr le 1

Inminus1

simradic2eminus12

a2ix2i dσ

Snminus1kε

a2ix2i dσ

a2i +nsum

which imply

supaiisinA

Snminus1(1)

a2ix2i dσ

nminus1(x)

aiisinA

infinsum

a2i + supaiisinA

infinsum

limjrarrinfin

n(j)sum

(aij minus ai)2 = 0

Then we have

(1) ǫn(j)

(2) σn(j)minus1

j rarr infin

In particular En(j)

as j rarr infin

lim supjrarrinfin

W2(σn(j)minus1

γn(j)

a2iji)

jrarrinfin

infinsum

a2ij =radic2eminus1

infinsum

W2(γn(j)

a2iji γinfina2i

)2 =infinsum

radicn(j)minus1 bij

It holds that

W2(σn(j)minus1

δo)2 =

Sn(j)minus1

n(j)minus1 biji

y2 dσn(j)minus1

Snminus1(1)

n(j)sum

n(j)minus1(x)

n(j)sum

(n(j)minus 1)

Snminus1(1)

limjrarrinfin

(n(j)minus 1)

Snminus1(1)

We therefore have

limjrarrinfin

W2(σn(j)minus1

δo)2 = lim

jrarrinfin

n(j)sum

b2ij gt 0

radicn(j)minus1 bij

thatlimjrarrinfin

dP(ǫn(j)

radicn(j)minus1 bij

γn(j)

b2ij) = 0

We also have

limjrarrinfin

W2(γn(j)

b2ij δo)

2 = limjrarrinfin

jrarrinfin

n(j)sum

b2ij gt 0

limjrarrinfin

lim infjrarrinfin

n(j)sum

a2ij gt 0(ii)

n(j)sum

and bij =aij

maxAj 1

so that En(j)

lim infjrarrinfin

n(j)sum

b2ij gt 0 and

n(j)sum

b2ij le 1

fj En(j)

radicn(j)minus1 bij

Since en(j)

we see that en(j)

(minusBj) = en(j)

is box

is not an

(43) limjrarrinfin

n(j)sum

a2ij gt

infinsum

n(j)sum

a2ij minusinfinsum

Setting

aijk =

akj if i le k

aij if i ge k + 1

we have

limjrarrinfin

n(j)sum

a2ijk +

n(j)sum

a2ij minusksum

a2i gt ε0

n(j(k))sum

bullsumn(j(k))

References

1 Introduction

2 Preliminaries





25 Pyramid

26 Dissipation


28 Gaussian space



References

dϕEx(v)2 = t2g2 +

r2+O(θminus2Nε)

= t2(partR

+ (1minus t2)R2

r2+O(θminus2Nε)

limnrarrinfin

enradicnminus1ai(D

nNε) = 1(1)

limnrarrinfin

γna2i (Dn

nNε) = σnminus1

radicnminus1 ai

(DnNε) =

γna2i (Dn

ϕE if (En

radicnminus1ai e

nradicnminus1ai)

radicnminus1 ai σ

NεcapFn

|DnNε

capFnθ) and so

radicnminus1ai|Dn

Nε en

|DnNε

capFnθ γna2i

) rarr 0

PΓinfina2i

as nrarr infin

radicn(j)minus1 ai

Since En(j)

radicn(j)minus1 ai

radicnminus1 ai for

dropping

i dominates Γka2i

and so PΓinfina2i

supPΓinfin

a2i We thus obtain

which is a contra-

converges

The ellipsoid En(j)

converges

bεi =

ai if i le I(ε)

n(j)aiji ≺ E

n(j)(1+2ε)bεi = (1 + 2ε)E

PEn(j)

bεi =

2ε if i ge I(ε)

En(j)aiji ≺ E

in PΓinfinb2εi

I(ε)minus1

(1+ε)2a2i

and ΓkminusI(ε)+1

ρ(PΓkb2εiPΓ

I(ε)minus1

kb2εi

ΓI(ε)minus1

(1+ε)2a2i ) le τ(ε)

ρ(PΓinfinb2εi

PΓI(ε)minus1

(1+ε)2a2i) le τ(ε)

b2ε(l)i

I(ε(l))minus1

infin as

as l rarr infin

PΓinfina2i

does not converge

PEn(j)

i as j rarr infin

asymptotically

a2 is not

i contains PΓinfin

i is not

a2ij we have

ObsDiam(Γn(j)

a2ijconverges

a2i as j rarr

converges

so that

and so PΓn(j)

a2ijto PΓinfin

converges

such that

supaiisinA

infinsum

a2i lt +infin

Then we have

limnrarrinfin

supaiisinA

) = 0(1)

lim supnrarrinfin

supaiisinA

aiisinA

infinsum

a2i(2)

radicnminus1

radicnminus1 ])

radicnminus 1 )

We remark that

radicnminus 1 ]))

limnrarrinfin

wθn = 1

It then holds that

nθ ) le dTV(ǫ

nradicnminus1 ai ǫ

dP(γna2i

ψ(x) =R

radicnminus 1 ])

W2(ǫnθ γ

2 leint

rminus 1

x2 dγnθ (x)

radicnminus 1 ]))

x2 dγna2i (x)

and hence

lim supnrarrinfin

supaiisinA

aiisinA

infinsum

This proves (1)

radicnminus1ai

)2 leint

Snminus1(1)

a2ix2i dσ

nminus1(x) middot 1

Inminus1

int infin

Inminus1

int infin

22 dr le 1

Inminus1

simradic2eminus12

a2ix2i dσ

Snminus1kε

a2ix2i dσ

a2i +nsum

which imply

supaiisinA

Snminus1(1)

a2ix2i dσ

nminus1(x)

aiisinA

infinsum

a2i + supaiisinA

infinsum

limjrarrinfin

n(j)sum

(aij minus ai)2 = 0

Then we have

(1) ǫn(j)

(2) σn(j)minus1

j rarr infin

In particular En(j)

as j rarr infin

lim supjrarrinfin

W2(σn(j)minus1

γn(j)

a2iji)

jrarrinfin

infinsum

a2ij =radic2eminus1

infinsum

W2(γn(j)

a2iji γinfina2i

)2 =infinsum

radicn(j)minus1 bij

It holds that

W2(σn(j)minus1

δo)2 =

Sn(j)minus1

n(j)minus1 biji

y2 dσn(j)minus1

Snminus1(1)

n(j)sum

n(j)minus1(x)

n(j)sum

(n(j)minus 1)

Snminus1(1)

limjrarrinfin

(n(j)minus 1)

Snminus1(1)

We therefore have

limjrarrinfin

W2(σn(j)minus1

δo)2 = lim

jrarrinfin

n(j)sum

b2ij gt 0

radicn(j)minus1 bij

thatlimjrarrinfin

dP(ǫn(j)

radicn(j)minus1 bij

γn(j)

b2ij) = 0

We also have

limjrarrinfin

W2(γn(j)

b2ij δo)

2 = limjrarrinfin

jrarrinfin

n(j)sum

b2ij gt 0

limjrarrinfin

lim infjrarrinfin

n(j)sum

a2ij gt 0(ii)

n(j)sum

and bij =aij

maxAj 1

so that En(j)

lim infjrarrinfin

n(j)sum

b2ij gt 0 and

n(j)sum

b2ij le 1

fj En(j)

radicn(j)minus1 bij

Since en(j)

we see that en(j)

(minusBj) = en(j)

is box

is not an

(43) limjrarrinfin

n(j)sum

a2ij gt

infinsum

n(j)sum

a2ij minusinfinsum

Setting

aijk =

akj if i le k

aij if i ge k + 1

we have

limjrarrinfin

n(j)sum

a2ijk +

n(j)sum

a2ij minusksum

a2i gt ε0

n(j(k))sum

bullsumn(j(k))

References

1 Introduction

2 Preliminaries





25 Pyramid

26 Dissipation


28 Gaussian space



References

radicnminus1ai|Dn

Nε en

|DnNε

capFnθ γna2i

) rarr 0

PΓinfina2i

as nrarr infin

radicn(j)minus1 ai

Since En(j)

radicn(j)minus1 ai

radicnminus1 ai for

dropping

i dominates Γka2i

and so PΓinfina2i

supPΓinfin

a2i We thus obtain

which is a contra-

converges

The ellipsoid En(j)

converges

bεi =

ai if i le I(ε)

n(j)aiji ≺ E

n(j)(1+2ε)bεi = (1 + 2ε)E

PEn(j)

bεi =

2ε if i ge I(ε)

En(j)aiji ≺ E

in PΓinfinb2εi

I(ε)minus1

(1+ε)2a2i

and ΓkminusI(ε)+1

ρ(PΓkb2εiPΓ

I(ε)minus1

kb2εi

ΓI(ε)minus1

(1+ε)2a2i ) le τ(ε)

ρ(PΓinfinb2εi

PΓI(ε)minus1

(1+ε)2a2i) le τ(ε)

b2ε(l)i

I(ε(l))minus1

infin as

as l rarr infin

PΓinfina2i

does not converge

PEn(j)

i as j rarr infin

asymptotically

a2 is not

i contains PΓinfin

i is not

a2ij we have

ObsDiam(Γn(j)

a2ijconverges

a2i as j rarr

converges

so that

and so PΓn(j)

a2ijto PΓinfin

converges

such that

supaiisinA

infinsum

a2i lt +infin

Then we have

limnrarrinfin

supaiisinA

) = 0(1)

lim supnrarrinfin

supaiisinA

aiisinA

infinsum

a2i(2)

radicnminus1

radicnminus1 ])

radicnminus 1 )

We remark that

radicnminus 1 ]))

limnrarrinfin

wθn = 1

It then holds that

nθ ) le dTV(ǫ

nradicnminus1 ai ǫ

dP(γna2i

ψ(x) =R

radicnminus 1 ])

W2(ǫnθ γ

2 leint

rminus 1

x2 dγnθ (x)

radicnminus 1 ]))

x2 dγna2i (x)

and hence

lim supnrarrinfin

supaiisinA

aiisinA

infinsum

This proves (1)

radicnminus1ai

)2 leint

Snminus1(1)

a2ix2i dσ

nminus1(x) middot 1

Inminus1

int infin

Inminus1

int infin

22 dr le 1

Inminus1

simradic2eminus12

a2ix2i dσ

Snminus1kε

a2ix2i dσ

a2i +nsum

which imply

supaiisinA

Snminus1(1)

a2ix2i dσ

nminus1(x)

aiisinA

infinsum

a2i + supaiisinA

infinsum

limjrarrinfin

n(j)sum

(aij minus ai)2 = 0

Then we have

(1) ǫn(j)

(2) σn(j)minus1

j rarr infin

In particular En(j)

as j rarr infin

lim supjrarrinfin

W2(σn(j)minus1

γn(j)

a2iji)

jrarrinfin

infinsum

a2ij =radic2eminus1

infinsum

W2(γn(j)

a2iji γinfina2i

)2 =infinsum

radicn(j)minus1 bij

It holds that

W2(σn(j)minus1

δo)2 =

Sn(j)minus1

n(j)minus1 biji

y2 dσn(j)minus1

Snminus1(1)

n(j)sum

n(j)minus1(x)

n(j)sum

(n(j)minus 1)

Snminus1(1)

limjrarrinfin

(n(j)minus 1)

Snminus1(1)

We therefore have

limjrarrinfin

W2(σn(j)minus1

δo)2 = lim

jrarrinfin

n(j)sum

b2ij gt 0

radicn(j)minus1 bij

thatlimjrarrinfin

dP(ǫn(j)

radicn(j)minus1 bij

γn(j)

b2ij) = 0

We also have

limjrarrinfin

W2(γn(j)

b2ij δo)

2 = limjrarrinfin

jrarrinfin

n(j)sum

b2ij gt 0

limjrarrinfin

lim infjrarrinfin

n(j)sum

a2ij gt 0(ii)

n(j)sum

and bij =aij

maxAj 1

so that En(j)

lim infjrarrinfin

n(j)sum

b2ij gt 0 and

n(j)sum

b2ij le 1

fj En(j)

radicn(j)minus1 bij

Since en(j)

we see that en(j)

(minusBj) = en(j)

is box

is not an

(43) limjrarrinfin

n(j)sum

a2ij gt

infinsum

n(j)sum

a2ij minusinfinsum

Setting

aijk =

akj if i le k

aij if i ge k + 1

we have

limjrarrinfin

n(j)sum

a2ijk +

n(j)sum

a2ij minusksum

a2i gt ε0

n(j(k))sum

bullsumn(j(k))

References

1 Introduction

2 Preliminaries





25 Pyramid

26 Dissipation


28 Gaussian space



References

The ellipsoid En(j)

converges

bεi =

ai if i le I(ε)

n(j)aiji ≺ E

n(j)(1+2ε)bεi = (1 + 2ε)E

PEn(j)

bεi =

2ε if i ge I(ε)

En(j)aiji ≺ E

in PΓinfinb2εi

I(ε)minus1

(1+ε)2a2i

and ΓkminusI(ε)+1

ρ(PΓkb2εiPΓ

I(ε)minus1

kb2εi

ΓI(ε)minus1

(1+ε)2a2i ) le τ(ε)

ρ(PΓinfinb2εi

PΓI(ε)minus1

(1+ε)2a2i) le τ(ε)

b2ε(l)i

I(ε(l))minus1

infin as

as l rarr infin

PΓinfina2i

does not converge

PEn(j)

i as j rarr infin

asymptotically

a2 is not

i contains PΓinfin

i is not

a2ij we have

ObsDiam(Γn(j)

a2ijconverges

a2i as j rarr

converges

so that

and so PΓn(j)

a2ijto PΓinfin

converges

such that

supaiisinA

infinsum

a2i lt +infin

Then we have

limnrarrinfin

supaiisinA

) = 0(1)

lim supnrarrinfin

supaiisinA

aiisinA

infinsum

a2i(2)

radicnminus1

radicnminus1 ])

radicnminus 1 )

We remark that

radicnminus 1 ]))

limnrarrinfin

wθn = 1

It then holds that

nθ ) le dTV(ǫ

nradicnminus1 ai ǫ

dP(γna2i

ψ(x) =R

radicnminus 1 ])

W2(ǫnθ γ

2 leint

rminus 1

x2 dγnθ (x)

radicnminus 1 ]))

x2 dγna2i (x)

and hence

lim supnrarrinfin

supaiisinA

aiisinA

infinsum

This proves (1)

radicnminus1ai

)2 leint

Snminus1(1)

a2ix2i dσ

nminus1(x) middot 1

Inminus1

int infin

Inminus1

int infin

22 dr le 1

Inminus1

simradic2eminus12

a2ix2i dσ

Snminus1kε

a2ix2i dσ

a2i +nsum

which imply

supaiisinA

Snminus1(1)

a2ix2i dσ

nminus1(x)

aiisinA

infinsum

a2i + supaiisinA

infinsum

limjrarrinfin

n(j)sum

(aij minus ai)2 = 0

Then we have

(1) ǫn(j)

(2) σn(j)minus1

j rarr infin

In particular En(j)

as j rarr infin

lim supjrarrinfin

W2(σn(j)minus1

γn(j)

a2iji)

jrarrinfin

infinsum

a2ij =radic2eminus1

infinsum

W2(γn(j)

a2iji γinfina2i

)2 =infinsum

radicn(j)minus1 bij

It holds that

W2(σn(j)minus1

δo)2 =

Sn(j)minus1

n(j)minus1 biji

y2 dσn(j)minus1

Snminus1(1)

n(j)sum

n(j)minus1(x)

n(j)sum

(n(j)minus 1)

Snminus1(1)

limjrarrinfin

(n(j)minus 1)

Snminus1(1)

We therefore have

limjrarrinfin

W2(σn(j)minus1

δo)2 = lim

jrarrinfin

n(j)sum

b2ij gt 0

radicn(j)minus1 bij

thatlimjrarrinfin

dP(ǫn(j)

radicn(j)minus1 bij

γn(j)

b2ij) = 0

We also have

limjrarrinfin

W2(γn(j)

b2ij δo)

2 = limjrarrinfin

jrarrinfin

n(j)sum

b2ij gt 0

limjrarrinfin

lim infjrarrinfin

n(j)sum

a2ij gt 0(ii)

n(j)sum

and bij =aij

maxAj 1

so that En(j)

lim infjrarrinfin

n(j)sum

b2ij gt 0 and

n(j)sum

b2ij le 1

fj En(j)

radicn(j)minus1 bij

Since en(j)

we see that en(j)

(minusBj) = en(j)

is box

is not an

(43) limjrarrinfin

n(j)sum

a2ij gt

infinsum

n(j)sum

a2ij minusinfinsum

Setting

aijk =

akj if i le k

aij if i ge k + 1

we have

limjrarrinfin

n(j)sum

a2ijk +

n(j)sum

a2ij minusksum

a2i gt ε0

n(j(k))sum

bullsumn(j(k))

References

1 Introduction

2 Preliminaries





25 Pyramid

26 Dissipation


28 Gaussian space



References

bεi =

2ε if i ge I(ε)

En(j)aiji ≺ E

in PΓinfinb2εi

I(ε)minus1

(1+ε)2a2i

and ΓkminusI(ε)+1

ρ(PΓkb2εiPΓ

I(ε)minus1

kb2εi

ΓI(ε)minus1

(1+ε)2a2i ) le τ(ε)

ρ(PΓinfinb2εi

PΓI(ε)minus1

(1+ε)2a2i) le τ(ε)

b2ε(l)i

I(ε(l))minus1

infin as

as l rarr infin

PΓinfina2i

does not converge

PEn(j)

i as j rarr infin

asymptotically

a2 is not

i contains PΓinfin

i is not

a2ij we have

ObsDiam(Γn(j)

a2ijconverges

a2i as j rarr

converges

so that

and so PΓn(j)

a2ijto PΓinfin

converges

such that

supaiisinA

infinsum

a2i lt +infin

Then we have

limnrarrinfin

supaiisinA

) = 0(1)

lim supnrarrinfin

supaiisinA

aiisinA

infinsum

a2i(2)

radicnminus1

radicnminus1 ])

radicnminus 1 )

We remark that

radicnminus 1 ]))

limnrarrinfin

wθn = 1

It then holds that

nθ ) le dTV(ǫ

nradicnminus1 ai ǫ

dP(γna2i

ψ(x) =R

radicnminus 1 ])

W2(ǫnθ γ

2 leint

rminus 1

x2 dγnθ (x)

radicnminus 1 ]))

x2 dγna2i (x)

and hence

lim supnrarrinfin

supaiisinA

aiisinA

infinsum

This proves (1)

radicnminus1ai

)2 leint

Snminus1(1)

a2ix2i dσ

nminus1(x) middot 1

Inminus1

int infin

Inminus1

int infin

22 dr le 1

Inminus1

simradic2eminus12

a2ix2i dσ

Snminus1kε

a2ix2i dσ

a2i +nsum

which imply

supaiisinA

Snminus1(1)

a2ix2i dσ

nminus1(x)

aiisinA

infinsum

a2i + supaiisinA

infinsum

limjrarrinfin

n(j)sum

(aij minus ai)2 = 0

Then we have

(1) ǫn(j)

(2) σn(j)minus1

j rarr infin

In particular En(j)

as j rarr infin

lim supjrarrinfin

W2(σn(j)minus1

γn(j)

a2iji)

jrarrinfin

infinsum

a2ij =radic2eminus1

infinsum

W2(γn(j)

a2iji γinfina2i

)2 =infinsum

radicn(j)minus1 bij

It holds that

W2(σn(j)minus1

δo)2 =

Sn(j)minus1

n(j)minus1 biji

y2 dσn(j)minus1

Snminus1(1)

n(j)sum

n(j)minus1(x)

n(j)sum

(n(j)minus 1)

Snminus1(1)

limjrarrinfin

(n(j)minus 1)

Snminus1(1)

We therefore have

limjrarrinfin

W2(σn(j)minus1

δo)2 = lim

jrarrinfin

n(j)sum

b2ij gt 0

radicn(j)minus1 bij

thatlimjrarrinfin

dP(ǫn(j)

radicn(j)minus1 bij

γn(j)

b2ij) = 0

We also have

limjrarrinfin

W2(γn(j)

b2ij δo)

2 = limjrarrinfin

jrarrinfin

n(j)sum

b2ij gt 0

limjrarrinfin

lim infjrarrinfin

n(j)sum

a2ij gt 0(ii)

n(j)sum

and bij =aij

maxAj 1

so that En(j)

lim infjrarrinfin

n(j)sum

b2ij gt 0 and

n(j)sum

b2ij le 1

fj En(j)

radicn(j)minus1 bij

Since en(j)

we see that en(j)

(minusBj) = en(j)

is box

is not an

(43) limjrarrinfin

n(j)sum

a2ij gt

infinsum

n(j)sum

a2ij minusinfinsum

Setting

aijk =

akj if i le k

aij if i ge k + 1

we have

limjrarrinfin

n(j)sum

a2ijk +

n(j)sum

a2ij minusksum

a2i gt ε0

n(j(k))sum

bullsumn(j(k))

References

1 Introduction

2 Preliminaries





25 Pyramid

26 Dissipation


28 Gaussian space



References

asymptotically

a2 is not

i contains PΓinfin

i is not

a2ij we have

ObsDiam(Γn(j)

a2ijconverges

a2i as j rarr

converges

so that

and so PΓn(j)

a2ijto PΓinfin

converges

such that

supaiisinA

infinsum

a2i lt +infin

Then we have

limnrarrinfin

supaiisinA

) = 0(1)

lim supnrarrinfin

supaiisinA

aiisinA

infinsum

a2i(2)

radicnminus1

radicnminus1 ])

radicnminus 1 )

We remark that

radicnminus 1 ]))

limnrarrinfin

wθn = 1

It then holds that

nθ ) le dTV(ǫ

nradicnminus1 ai ǫ

dP(γna2i

ψ(x) =R

radicnminus 1 ])

W2(ǫnθ γ

2 leint

rminus 1

x2 dγnθ (x)

radicnminus 1 ]))

x2 dγna2i (x)

and hence

lim supnrarrinfin

supaiisinA

aiisinA

infinsum

This proves (1)

radicnminus1ai

)2 leint

Snminus1(1)

a2ix2i dσ

nminus1(x) middot 1

Inminus1

int infin

Inminus1

int infin

22 dr le 1

Inminus1

simradic2eminus12

a2ix2i dσ

Snminus1kε

a2ix2i dσ

a2i +nsum

which imply

supaiisinA

Snminus1(1)

a2ix2i dσ

nminus1(x)

aiisinA

infinsum

a2i + supaiisinA

infinsum

limjrarrinfin

n(j)sum

(aij minus ai)2 = 0

Then we have

(1) ǫn(j)

(2) σn(j)minus1

j rarr infin

In particular En(j)

as j rarr infin

lim supjrarrinfin

W2(σn(j)minus1

γn(j)

a2iji)

jrarrinfin

infinsum

a2ij =radic2eminus1

infinsum

W2(γn(j)

a2iji γinfina2i

)2 =infinsum

radicn(j)minus1 bij

It holds that

W2(σn(j)minus1

δo)2 =

Sn(j)minus1

n(j)minus1 biji

y2 dσn(j)minus1

Snminus1(1)

n(j)sum

n(j)minus1(x)

n(j)sum

(n(j)minus 1)

Snminus1(1)

limjrarrinfin

(n(j)minus 1)

Snminus1(1)

We therefore have

limjrarrinfin

W2(σn(j)minus1

δo)2 = lim

jrarrinfin

n(j)sum

b2ij gt 0

radicn(j)minus1 bij

thatlimjrarrinfin

dP(ǫn(j)

radicn(j)minus1 bij

γn(j)

b2ij) = 0

We also have

limjrarrinfin

W2(γn(j)

b2ij δo)

2 = limjrarrinfin

jrarrinfin

n(j)sum

b2ij gt 0

limjrarrinfin

lim infjrarrinfin

n(j)sum

a2ij gt 0(ii)

n(j)sum

and bij =aij

maxAj 1

so that En(j)

lim infjrarrinfin

n(j)sum

b2ij gt 0 and

n(j)sum

b2ij le 1

fj En(j)

radicn(j)minus1 bij

Since en(j)

we see that en(j)

(minusBj) = en(j)

is box

is not an

(43) limjrarrinfin

n(j)sum

a2ij gt

infinsum

n(j)sum

a2ij minusinfinsum

Setting

aijk =

akj if i le k

aij if i ge k + 1

we have

limjrarrinfin

n(j)sum

a2ijk +

n(j)sum

a2ij minusksum

a2i gt ε0

n(j(k))sum

bullsumn(j(k))

References

1 Introduction

2 Preliminaries





25 Pyramid

26 Dissipation


28 Gaussian space



References

so that

and so PΓn(j)

a2ijto PΓinfin

converges

such that

supaiisinA

infinsum

a2i lt +infin

Then we have

limnrarrinfin

supaiisinA

) = 0(1)

lim supnrarrinfin

supaiisinA

aiisinA

infinsum

a2i(2)

radicnminus1

radicnminus1 ])

radicnminus 1 )

We remark that

radicnminus 1 ]))

limnrarrinfin

wθn = 1

It then holds that

nθ ) le dTV(ǫ

nradicnminus1 ai ǫ

dP(γna2i

ψ(x) =R

radicnminus 1 ])

W2(ǫnθ γ

2 leint

rminus 1

x2 dγnθ (x)

radicnminus 1 ]))

x2 dγna2i (x)

and hence

lim supnrarrinfin

supaiisinA

aiisinA

infinsum

This proves (1)

radicnminus1ai

)2 leint

Snminus1(1)

a2ix2i dσ

nminus1(x) middot 1

Inminus1

int infin

Inminus1

int infin

22 dr le 1

Inminus1

simradic2eminus12

a2ix2i dσ

Snminus1kε

a2ix2i dσ

a2i +nsum

which imply

supaiisinA

Snminus1(1)

a2ix2i dσ

nminus1(x)

aiisinA

infinsum

a2i + supaiisinA

infinsum

limjrarrinfin

n(j)sum

(aij minus ai)2 = 0

Then we have

(1) ǫn(j)

(2) σn(j)minus1

j rarr infin

In particular En(j)

as j rarr infin

lim supjrarrinfin

W2(σn(j)minus1

γn(j)

a2iji)

jrarrinfin

infinsum

a2ij =radic2eminus1

infinsum

W2(γn(j)

a2iji γinfina2i

)2 =infinsum

radicn(j)minus1 bij

It holds that

W2(σn(j)minus1

δo)2 =

Sn(j)minus1

n(j)minus1 biji

y2 dσn(j)minus1

Snminus1(1)

n(j)sum

n(j)minus1(x)

n(j)sum

(n(j)minus 1)

Snminus1(1)

limjrarrinfin

(n(j)minus 1)

Snminus1(1)

We therefore have

limjrarrinfin

W2(σn(j)minus1

δo)2 = lim

jrarrinfin

n(j)sum

b2ij gt 0

radicn(j)minus1 bij

thatlimjrarrinfin

dP(ǫn(j)

radicn(j)minus1 bij

γn(j)

b2ij) = 0

We also have

limjrarrinfin

W2(γn(j)

b2ij δo)

2 = limjrarrinfin

jrarrinfin

n(j)sum

b2ij gt 0

limjrarrinfin

lim infjrarrinfin

n(j)sum

a2ij gt 0(ii)

n(j)sum

and bij =aij

maxAj 1

so that En(j)

lim infjrarrinfin

n(j)sum

b2ij gt 0 and

n(j)sum

b2ij le 1

fj En(j)

radicn(j)minus1 bij

Since en(j)

we see that en(j)

(minusBj) = en(j)

is box

is not an

(43) limjrarrinfin

n(j)sum

a2ij gt

infinsum

n(j)sum

a2ij minusinfinsum

Setting

aijk =

akj if i le k

aij if i ge k + 1

we have

limjrarrinfin

n(j)sum

a2ijk +

n(j)sum

a2ij minusksum

a2i gt ε0

n(j(k))sum

bullsumn(j(k))

References

1 Introduction

2 Preliminaries





25 Pyramid

26 Dissipation


28 Gaussian space



References

We remark that

radicnminus 1 ]))

limnrarrinfin

wθn = 1

It then holds that

nθ ) le dTV(ǫ

nradicnminus1 ai ǫ

dP(γna2i

ψ(x) =R

radicnminus 1 ])

W2(ǫnθ γ

2 leint

rminus 1

x2 dγnθ (x)

radicnminus 1 ]))

x2 dγna2i (x)

and hence

lim supnrarrinfin

supaiisinA

aiisinA

infinsum

This proves (1)

radicnminus1ai

)2 leint

Snminus1(1)

a2ix2i dσ

nminus1(x) middot 1

Inminus1

int infin

Inminus1

int infin

22 dr le 1

Inminus1

simradic2eminus12

a2ix2i dσ

Snminus1kε

a2ix2i dσ

a2i +nsum

which imply

supaiisinA

Snminus1(1)

a2ix2i dσ

nminus1(x)

aiisinA

infinsum

a2i + supaiisinA

infinsum

limjrarrinfin

n(j)sum

(aij minus ai)2 = 0

Then we have

(1) ǫn(j)

(2) σn(j)minus1

j rarr infin

In particular En(j)

as j rarr infin

lim supjrarrinfin

W2(σn(j)minus1

γn(j)

a2iji)

jrarrinfin

infinsum

a2ij =radic2eminus1

infinsum

W2(γn(j)

a2iji γinfina2i

)2 =infinsum

radicn(j)minus1 bij

It holds that

W2(σn(j)minus1

δo)2 =

Sn(j)minus1

n(j)minus1 biji

y2 dσn(j)minus1

Snminus1(1)

n(j)sum

n(j)minus1(x)

n(j)sum

(n(j)minus 1)

Snminus1(1)

limjrarrinfin

(n(j)minus 1)

Snminus1(1)

We therefore have

limjrarrinfin

W2(σn(j)minus1

δo)2 = lim

jrarrinfin

n(j)sum

b2ij gt 0

radicn(j)minus1 bij

thatlimjrarrinfin

dP(ǫn(j)

radicn(j)minus1 bij

γn(j)

b2ij) = 0

We also have

limjrarrinfin

W2(γn(j)

b2ij δo)

2 = limjrarrinfin

jrarrinfin

n(j)sum

b2ij gt 0

limjrarrinfin

lim infjrarrinfin

n(j)sum

a2ij gt 0(ii)

n(j)sum

and bij =aij

maxAj 1

so that En(j)

lim infjrarrinfin

n(j)sum

b2ij gt 0 and

n(j)sum

b2ij le 1

fj En(j)

radicn(j)minus1 bij

Since en(j)

we see that en(j)

(minusBj) = en(j)

is box

is not an

(43) limjrarrinfin

n(j)sum

a2ij gt

infinsum

n(j)sum

a2ij minusinfinsum

Setting

aijk =

akj if i le k

aij if i ge k + 1

we have

limjrarrinfin

n(j)sum

a2ijk +

n(j)sum

a2ij minusksum

a2i gt ε0

n(j(k))sum

bullsumn(j(k))

References

1 Introduction

2 Preliminaries





25 Pyramid

26 Dissipation


28 Gaussian space



References

radicnminus1ai

)2 leint

Snminus1(1)

a2ix2i dσ

nminus1(x) middot 1

Inminus1

int infin

Inminus1

int infin

22 dr le 1

Inminus1

simradic2eminus12

a2ix2i dσ

Snminus1kε

a2ix2i dσ

a2i +nsum

which imply

supaiisinA

Snminus1(1)

a2ix2i dσ

nminus1(x)

aiisinA

infinsum

a2i + supaiisinA

infinsum

limjrarrinfin

n(j)sum

(aij minus ai)2 = 0

Then we have

(1) ǫn(j)

(2) σn(j)minus1

j rarr infin

In particular En(j)

as j rarr infin

lim supjrarrinfin

W2(σn(j)minus1

γn(j)

a2iji)

jrarrinfin

infinsum

a2ij =radic2eminus1

infinsum

W2(γn(j)

a2iji γinfina2i

)2 =infinsum

radicn(j)minus1 bij

It holds that

W2(σn(j)minus1

δo)2 =

Sn(j)minus1

n(j)minus1 biji

y2 dσn(j)minus1

Snminus1(1)

n(j)sum

n(j)minus1(x)

n(j)sum

(n(j)minus 1)

Snminus1(1)

limjrarrinfin

(n(j)minus 1)

Snminus1(1)

We therefore have

limjrarrinfin

W2(σn(j)minus1

δo)2 = lim

jrarrinfin

n(j)sum

b2ij gt 0

radicn(j)minus1 bij

thatlimjrarrinfin

dP(ǫn(j)

radicn(j)minus1 bij

γn(j)

b2ij) = 0

We also have

limjrarrinfin

W2(γn(j)

b2ij δo)

2 = limjrarrinfin

jrarrinfin

n(j)sum

b2ij gt 0

limjrarrinfin

lim infjrarrinfin

n(j)sum

a2ij gt 0(ii)

n(j)sum

and bij =aij

maxAj 1

so that En(j)

lim infjrarrinfin

n(j)sum

b2ij gt 0 and

n(j)sum

b2ij le 1

fj En(j)

radicn(j)minus1 bij

Since en(j)

we see that en(j)

(minusBj) = en(j)

is box

is not an

(43) limjrarrinfin

n(j)sum

a2ij gt

infinsum

n(j)sum

a2ij minusinfinsum

Setting

aijk =

akj if i le k

aij if i ge k + 1

we have

limjrarrinfin

n(j)sum

a2ijk +

n(j)sum

a2ij minusksum

a2i gt ε0

n(j(k))sum

bullsumn(j(k))

References

1 Introduction

2 Preliminaries





25 Pyramid

26 Dissipation


28 Gaussian space



References

(2) σn(j)minus1

j rarr infin

In particular En(j)

as j rarr infin

lim supjrarrinfin

W2(σn(j)minus1

γn(j)

a2iji)

jrarrinfin

infinsum

a2ij =radic2eminus1

infinsum

W2(γn(j)

a2iji γinfina2i

)2 =infinsum

radicn(j)minus1 bij

It holds that

W2(σn(j)minus1

δo)2 =

Sn(j)minus1

n(j)minus1 biji

y2 dσn(j)minus1

Snminus1(1)

n(j)sum

n(j)minus1(x)

n(j)sum

(n(j)minus 1)

Snminus1(1)

limjrarrinfin

(n(j)minus 1)

Snminus1(1)

We therefore have

limjrarrinfin

W2(σn(j)minus1

δo)2 = lim

jrarrinfin

n(j)sum

b2ij gt 0

radicn(j)minus1 bij

thatlimjrarrinfin

dP(ǫn(j)

radicn(j)minus1 bij

γn(j)

b2ij) = 0

We also have

limjrarrinfin

W2(γn(j)

b2ij δo)

2 = limjrarrinfin

jrarrinfin

n(j)sum

b2ij gt 0

limjrarrinfin

lim infjrarrinfin

n(j)sum

a2ij gt 0(ii)

n(j)sum

and bij =aij

maxAj 1

so that En(j)

lim infjrarrinfin

n(j)sum

b2ij gt 0 and

n(j)sum

b2ij le 1

fj En(j)

radicn(j)minus1 bij

Since en(j)

we see that en(j)

(minusBj) = en(j)

is box

is not an

(43) limjrarrinfin

n(j)sum

a2ij gt

infinsum

n(j)sum

a2ij minusinfinsum

Setting

aijk =

akj if i le k

aij if i ge k + 1

we have

limjrarrinfin

n(j)sum

a2ijk +

n(j)sum

a2ij minusksum

a2i gt ε0

n(j(k))sum

bullsumn(j(k))

References

1 Introduction

2 Preliminaries





25 Pyramid

26 Dissipation


28 Gaussian space



References

We therefore have

limjrarrinfin

W2(σn(j)minus1

δo)2 = lim

jrarrinfin

n(j)sum

b2ij gt 0

radicn(j)minus1 bij

thatlimjrarrinfin

dP(ǫn(j)

radicn(j)minus1 bij

γn(j)

b2ij) = 0

We also have

limjrarrinfin

W2(γn(j)

b2ij δo)

2 = limjrarrinfin

jrarrinfin

n(j)sum

b2ij gt 0

limjrarrinfin

lim infjrarrinfin

n(j)sum

a2ij gt 0(ii)

n(j)sum

and bij =aij

maxAj 1

so that En(j)

lim infjrarrinfin

n(j)sum

b2ij gt 0 and

n(j)sum

b2ij le 1

fj En(j)

radicn(j)minus1 bij

Since en(j)

we see that en(j)

(minusBj) = en(j)

is box

is not an

(43) limjrarrinfin

n(j)sum

a2ij gt

infinsum

n(j)sum

a2ij minusinfinsum

Setting

aijk =

akj if i le k

aij if i ge k + 1

we have

limjrarrinfin

n(j)sum

a2ijk +

n(j)sum

a2ij minusksum

a2i gt ε0

n(j(k))sum

bullsumn(j(k))

References

1 Introduction

2 Preliminaries





25 Pyramid

26 Dissipation


28 Gaussian space



References

lim infjrarrinfin

n(j)sum

b2ij gt 0 and

n(j)sum

b2ij le 1

fj En(j)

radicn(j)minus1 bij

Since en(j)

we see that en(j)

(minusBj) = en(j)

is box

is not an

(43) limjrarrinfin

n(j)sum

a2ij gt

infinsum

n(j)sum

a2ij minusinfinsum

Setting

aijk =

akj if i le k

aij if i ge k + 1

we have

limjrarrinfin

n(j)sum

a2ijk +

n(j)sum

a2ij minusksum

a2i gt ε0

n(j(k))sum

bullsumn(j(k))

References

1 Introduction

2 Preliminaries





25 Pyramid

26 Dissipation


28 Gaussian space



References

n(j)sum

a2ij minusinfinsum

Setting

aijk =

akj if i le k

aij if i ge k + 1

we have

limjrarrinfin

n(j)sum

a2ijk +

n(j)sum

a2ij minusksum

a2i gt ε0

n(j(k))sum

bullsumn(j(k))

References

1 Introduction

2 Preliminaries





25 Pyramid

26 Dissipation


28 Gaussian space



References

1 Introduction

2 Preliminaries





25 Pyramid

26 Dissipation


28 Gaussian space



References

1 Introduction

2 Preliminaries





25 Pyramid

26 Dissipation


28 Gaussian space



References

Documents

HIGH-DIMENSIONAL ELLIPSOIDS CONVERGE TO GAUSSIAN … · CONVERGE TO GAUSSIAN SPACES DAISUKE KAZUKAWA AND TAKASHI SHIOYA Abstract. We prove the convergence of (solid) ellipsoids to