Normal Approximation from a Stein Identity (Based …Normal Approximation from a Stein Identity (Based on joint work (in progress) with Aihua Xia) Louis H. Y. Chen National University

Normal Approximation from a Stein Identity(Based on joint work (in progress) with Aihua Xia)

Louis H. Y. Chen

National University of Singapore

Workshop on New Directions in Stein’s Method18 - 29 May 2015

Institute for Mathematical SciencesNational Universty of Singapore

L. H. Y. Chen (NUS) Normal approximation Stein’s method workshop 1 / 29

Outline

Motivation

A General Theorem

Sums of Independent Random Variables

Random Measures

Completely Random Measures

Locally Dependent Random Measures

Maximal Points

Excursion Random Measures

Ginibre-Voronoi Tessellation

Stein Couplings

Summary


Motivation

Many problems in geometric probability can be formulated interms of random measures.

There is a Palm measure associated with each random measure -process version of size-biasness.

If we can couple the Palm measure to the random measure, wecan construct a Stein identity.

In this case, can we obtain a general Kolmogorov bound in thenormal approximation for random measures?


Motivation

Random measures and Palm measures

Γ a locally compact separable metric space.

Ξ a random measure on Γ with finite mean mesaure Λ, that is,Λ(A) = EΞ(A) for A ∈ B(Γ) and Λ(Γ) <∞.

Ξα the Palm measure associated with Ξ at α ∈ Γ, that is,

E(∫

Γ

f(α,Ξ)Ξ(dα)

)= E

(∫Γ

f(α,Ξα)Λ(dα)

). (1)

for real-valued functions f(·, ·) for which the expectations exist.


Motivation

Stein identity for random measures

By taking f to be absolutely continuous from R to R, (13)implies

E|Ξ|f(|Ξ|) = E∫

Γ

f(|Ξα|)Λ(dα). (2)

Let λ = Λ(Γ) = E|Ξ|, B2 = Var(|Ξ|), and define

W =|Ξ| − λB

, Wα =|Ξα| − λ

B.


Motivation

Stein identity for random measures (continued)

Assume that Ξ and Ξα, α ∈ Γ, are defined on the sameprobability space, and let

∆α = Wα −W.

From (2), we obtain

EWf(W ) = E∫ ∞−∞

f ′(W + t)K(t)dt

where

K(t) =1

B

∫Γ

[I(∆α > t > 0)− I(∆α < t ≤ 0)]Λ(dα).


Motivation

A general problem

Let W be such that EW = 0 and Var(W ) = 1. Suppose thereexists a random function K(t) such that

EWf(W ) = E∫ ∞−∞

f ′(W + t)K(t)dt (3)

for all absolutely continuous functions f for which theexpectations exist.

Can we obtain a meaningful bound on the Kolmogorov distancebetween L(W ) and N (0, 1) without further assumption?

Recall

dK(L(W ),N (0, 1)) := supx∈R|P (W ≤ x)− P (Z ≤ x)|,

Z ∼ N (0, 1).


A General Theorem

Theorem 1Let W be such that EW = 0 and Var(W ) = 1. Suppose there existsa random function K(t) such that

EWf(W ) = E∫ ∞−∞

f ′(W + t)K(t)dt

for all absolutely continuous functions f for which the expectationsexist.

Let K(t) = Kin(t) + Kout(t) where Kin(t) = 0 for |t| > 1. DefineK(t) = EK(t), Kin(t) = EKin(t), and Kout(t) = EKout(t).

Then

dK(L(W ),N (0, 1)) ≤ 2r1 + 12r2 + 8r3 + 9r4 + 15r5.


A General Theorem

In Theorem 1,

r1 =

[E(∫|t|≤1

(Kin(t)−Kin(t))dt

)2] 1

2

,

r2 =

∫|t|≤1

|tKin(t)|dt,

r3 = E∫ ∞−∞|Kout(t)|dt,

r4 = E∫|t|≤1

(Kin(t)−Kin(t))2dt,

r5 =

[E∫|t|≤1

|t|(Kin(t)−Kin(t))2dt

] 12

.


A General Theorem

Idea of proof

Adapt the techniques used in the proof of Theorem 2.1 (forLD1) in Chen and Shao (2004), Ann. Probab.

Instead of using a concentration inequality with explicit bound(whose proof is complicated), we use this inequality:For a ≤ b, a, b ∈ R,

P (a ≤ W ≤ b) ≤ 2dK(L(W ),N (0, 1)) +b− a√

2π.

Also use this inequality which plays a crucial role:For a, b, θ > 0,

ab ≤ θa2

2+b2

2θ.


A General Theorem

Idea of proof (continued)

E∫|t|≤1

I(x− 0 ∨ t ≤ W ≤ x− 0 ∧ t+ ε)|Kin(t)−Kin(t)|dt

≤ θβ

2E∫|t|≤1

I(x− 0 ∨ t ≤ W ≤ x− 0 ∧ t+ ε)

2d+ 0.4|t|+ 0.4εdt

+1

2θβE∫|t|≤1

(2d+ 0.4|t|+ 0.4ε)(Kin(t)−Kin(t))2dt

≤ θβ +d+ 0.2ε

θβr4 +

0.2

θβr2

5

≤ θd+ 0.2θε+1

θr4 + (θ +

0.2

θ)r5

(by letting β = d+ 0.2ε+ r5).


Sums of Independent Random Variables

What does the bound in Theorem 1 look like in the case of a sum ofindependent random variables?

Let ξ1, · · · , ξn be independent random variables with Eξi = 0,Var(ξi) = σ2

i , and E|ξi| <∞.

Let B2 =∑n

i=1 σ2i and let W =

1

B

n∑i=1

ξi.

Applying Theorem 1,

dK(L(W ),N (0, 1)) ≤17√∑n

i=1 Eξ4i I(|ξi| ≤ B)

B2+

21∑n

i=1 E|ξi|3

B3.

If both∑n

i=1 Eξ4i I(|ξi| ≤ B) and

∑ni=1 E|ξi|3 are O(B2), such

as in the i.i.d. case, then the bound is O(B−1), which agrees

with the order of the Berry-Esseen boundC∑n

i=1 E|ξi|3

B3.


Random Measures

Γ a locally compact separable metric space.

Ξ a random measure on Γ with finite mean mesaure Λ, that is,Λ(A) = EΞ(A) for A ∈ B(Γ) and Λ(Γ) <∞.

Ξα the Palm measure associated with Ξ at α ∈ Γ, that is,

E(∫

Γ

f(α,Ξ)Ξ(dα)

)= E

(∫Γ

f(α,Ξα)Λ(dα)

).

for real-valued functions f(·, ·) for which the expectations exist.

Assume that Ξ and Ξα, α ∈ Γ, are defined on the sameprobability space, and let ∆α = Wα −W .


Random Measures

We have

EWf(W ) = E∫ ∞−∞

f ′(W + t)K(t)dt

where

K(t) =1

B

∫Γ

χ(α, t)Λ(dα),

χ(α, t) = I(∆α > t > 0)− I(∆α < t ≤ 0).

Define

Kin(t) =1

B

∫Γ

χ(α, t)I(|∆α| ≤ 1)Λ(dα);

Kout(t) =1

B

∫Γ

χ(α, t)I(|∆α| > 1)Λ(dα).

Apply the main theorem (Theorem 1).



A random measure Ξ on Γ is completely random ifΞ(A1), · · · ,Ξ(Ak) are independent wheneverA1, · · · , Ak ∈ B(Γ) are pairwise disjoint (Kingman (1967),Pacific J. Math.)

Examples are the compound Poisson process with clusterdistributions on R+ and the gamma process.

For the compound Poisson process and the gamma process,Γ = [0, T ].

These processes are not covered by the result of Barbour andXia (2006), Adv. Appl. Probab.



Theorem 2Let Ξ be a completely random measure with mean measure Λ andfinite second moment E|Ξ|2 <∞. Let µ := µΞ = Λ(Γ),B2 = Var(|Ξ|) and let W = (|Ξ| − E|Ξ|)/B. Then

dK(L(W ),N (0, 1)) ≤ 13B−2ε1/22,Ξ + 12B−3ε1,Ξ + 9B−3ε3,Ξ,

where

ε1,Ξ :=

∫Γ

E(Ξ({α})2 + Ξα({α})2)Λ(dα),

ε2,Ξ :=∑

Γ

Λ({α})2E(Ξ({α})2 + Ξα({α})2),

ε3,Ξ :=∑

Γ

Λ({α})2E(Ξ({α}) + Ξα({α})).



Corollary 3If the mean measure of the completely random measure is diffuse,that is, atomless, then

dK(L(W ),N (0, 1)) ≤ 12

B3ε1,Ξ.

Corollary 4

Let Ξ(i), 1 ≤ i ≤ n, be independent random measures on the carrier

space Γ and let Ξ =∑n

i=1 Ξ(i), B2 = Var(|Ξ|), W =|Ξ| − E|Ξ|

B.

Then

dK(L(W ),N (0, 1)) ≤ 19

B2

(n∑i=1

E|Ξ(i)|E|Ξ(i)|3)1/2

+42

B3

n∑i=1

E|Ξ(i)|3.



A random measure Ξ on Γ with mean measure Λ is locallydependent with dependency neighborhoods Bα, α ∈ Γ, if

L(Ξα|Bcα) = L(Ξ|Bcα) Λ− a.s.

(Chen and Xia (2004), Ann. Probab.)

An example: Matern hard-core process - produced from aPoisson point process by deleting any point within distance r ofanother point, irrespective of whether the latter has itselfalready deleted.

Can couple Ξα to Ξ such that ΞBcα = ΞBcα , α ∈ Γ.



Theorem 5Let Ξ be a random measure on Γ with finite mean measure Λ suchthat E|Ξ|4 <∞. Let B2 = Var(|Ξ|). Assume that Ξ and Ξα, α ∈ Γ,are defined on the same probability space. Define

W =|Ξ| − E|Ξ|

B, Wα =

|Ξα| − E|Ξ|B

, ∆α = Wα −W.

Assume that there is a set D ⊂ Γ× Γ such that for all (α, β) 6= D,∆α and ∆β are independent. Then

dK(L(W ),N (0, 1)) ≤ r1 + r2 + r3 + r4,

where



r1 =1

B

[∫(α,β)∈D

E|∆α∆β|Λ(dα)Λ(dβ)

] 12

;

r2 =1

B

∫Γ

E∆2αΛ(dα);

r3 =1

B2

∫(α,β)∈D

EMin(|∆α|, |∆β|)Λ(dα)Λ(dβ);

r4 =1

B

[∫(α,β)∈D

EMin(∆2α,∆

2β)Λ(dα)Λ(dβ)

] 12

.


Maximal Points

P a Poisson point process with rate µ in the region:

D := {α = (α1, α2) : 0 ≤ α1 ≤ 1, 0 ≤ α2 ≤ F (α1)},F (0) = 1, F (1) = 0, F ′(x) < 0,

bounded away from 0 and −∞.

A point α ∈ D ∩ P is maximal if P ∩ Aα = {α}.


Maximal Points

Define Ξ(dα) = I(P ∩ Aα = {α})P (dα).

Ξ is a random measure on D and |Ξ| counts the total number ofmaximal points of the Poisson point process P in D.

Ξ is locally dependent.

Theorem 6

Let W =|Ξ| − E|Ξ|√

Var(|Ξ|. We have

dK(L(W ),N (0, 1)) = O

(log µ

µ1/4

)Bound same order as in Barbour and Xia (2006), Adv. Appl.Probab., but proof simpler - uses local dependence of only order1, that is, LD1.


Maximal Points

Idea of proof


Excursion Random Measures

S a metric space.

{Xt : 0 ≤ t ≤ T} an l-dependent S-valued random process, thatis, {Xt : 0 ≤ t ≤ a} and {Xt : b ≤ t ≤ T} are independent ifb− a > l.

Define the excursion random measure

Ξ(dt) = I((t,Xt) ∈ E)dt, E ∈ B([0, T ]× S)

Theorem 7

Let µ = EΞ([0, t]), B2 = Var(|Ξ|) and W =|Ξ| − µB

. We have

dK(L(W ),N (0, 1)) = O

(l3/2µ1/2

B2+l2µ

B3

).



The Ginibre point process has attracted considerable attentionrecently because of its wide use in mobile networks and theGinibre-Voronoi tesselation.

The Ginibre point process is a special case of the Gibbs pointprocess family and it exhibits repulsion between points.

Goldman (2010), Ann. Appl. Probab. established that the Palmprocess Ξα associated with the Ginibre process Ξ at α ∈ Γsatisfies

L(Ξ) = L((Ξα \ {α}) ∪ {α + (Z1, Z2)}),

where

(Z1, Z2) ∼ N (0,1

2I2).



Voronoi diagram (20 points and their Voronoi cells )

Define η(dα) = (Total edge length of cell containing α)Ξ(dα).

η is a locally dependent random measure.L. H. Y. Chen (NUS) Normal approximation Stein’s method workshop 26 / 29

Stein Couplings

Let (W,W ′, G) be a Stein coupling with Var(W ) = 1.

E{Gf(W ′)−Gf(W )} = E{Wf(W )} for all f for which theexpectations exist.

Let ∆ = W ′ −W and let F be a σ-algebra w.r.t. which W ismeasurable.

EWf(W ) = E∫∞−∞ f

′(W + t)K(t)dt, where

K(t) = E{G[I(∆ > t > 0)− I(∆ < t ≤ 0)]∣∣F}.

Kin(t) = E{G[I(∆ > t > 0)− I(∆ < t ≤ 0)]I(|∆| ≤ 1)∣∣F}.


Summary

We gave motivation and presented a general theorem for normalapproximation in Kolmogorov distance based on a Stein identity.

The theorem is applied to random measures, which includecompletely random measures and locally dependent randommeasures.

Explicit bounds are obtained for the number of maximal pointsin a Poisson point process and the excursion random measure ofan l-dependent random process.

We discussed briefly the random measure of total edge length ofcells in a Ginibre-Voronoi tessellation.

We also discussed briefly the construction of a Stein identity forStein couplings.


Thank You


Documents

Normal Approximation from a Stein Identity (Based …Normal Approximation from a Stein Identity (Based on joint work (in progress) with Aihua Xia) Louis H. Y. Chen National University