Distributions with given Marginals and Moment Problems

Distributions with given Marginals and Moment Problems Edited by
Viktor BeneS Department of Mathematics, Czech Technical University, Prague, Czech Republic
and
SPRINGER SCIENCE+BUSINESS MEDIA, B.V.
A CLP. Catalogue record for this book is available from the Library of Congress.
ISBN 978-94-010-6329-6 ISBN 978-94-011-5532-8 (eBook) DOI 10.1007/978-94-011-5532-8
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All Rights Reserved © 1997 Springer Science+Business Media Dordrecht Originally published by Kluwer Academic Publishers in 1997 Softcover reprint of the hardcover 1st edition 1997
No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner.
TABLE OF CONTENTS
vii PREFACE ACKNOWLEDGEMENT x G. A. ANASTASSIOU / Optimal bounds on the average of a rounded
off observation in the presence of a single moment condition 1 T. RYCHLIK / The complete solution of a rounding problem under two
moment conditions 15 V. GIRARDIN / Methods of realization of moment problems with en-
tropy maximization 21 E. KAARIK / Matrices of higher moments: some problems of represen-
tation 27 L. B. KLEBANOV, S. T. RACHEV / The method of moments in to-
mography and quantum mechanics 35 V. BENES / Moment problems in stochastic geometry 53 M. SCARSINI, M. SHAKED / Frechet classes and nonmonotone de-
pendence 59 A. CHATEAUNEUF, M. COHEN, 1. MEILIJSON / Comonotonicity,
rank-dependent utilities and a search problem 73 P. CAPERAA, A. L. FOUGERES, C. GENEST / A stochastic ordering
based on a decomposition of Kendall's tau 81 M. J. W. JANSEN / Maximum entropy distributions with prescribed
marginals and normal score correlations 87 . V. E. PIPERIGOU / On bivariate distributions with Polya-Aeppli or L iiders-Delaporte marginals 93
E.-M. TIlT, H.-L. HELEMAE / Boundary distributions with fixed mar- ~~ ~
X. LI , P . MIKUSINSKI, H. SHERWOOD, M. D. TAYLOR / On approximations of copulas 107
G. DALL'AGLIO / Joint distributions of two uniform random variables when the sum and difference are independent 117
R. B. NELSEN, G. A. FREDRICKS / Diagonal copulas 121 G. A. FREDRICKS, R. B. NELSEN / Copulas constructed from diag-
onal sections 129 C. M. CUADRAS, J. FORTIANA / Continuous scaling on a bivariate
copula 137 H. G. KELLERER / Representation of Markov kernels by random map-
pings under order conditions 143 J. STEP AN / How to construct a two dimensional random vector with
a given conditional structure 161 A. HIRSHBERG, R. M. SHORTT / Strassen 's theorem for group-valued
charges 173
vi
G. LETAC / The Lancaster's probabilities on R2 and their extreme points 179
M. STUDENY / On marginalization, collapsibility and precollapsibili- ty 191
J. DUPACOV A / Moment bounds for stochastic programs in particular for recourse problems 199
T. SZANTAI / Probabilistic constrained programming and distributions with given marginals 205
V. KANKOV A / On an £-solution of minimax problem in stochastic programming 211
T. ViSEK / Bounds for stochastic programs - nonconvex case 217 R. JIROUSEK / Artificial intelligence, the marginal problem and in-
consistency 223 O. KIlii / Inconsistent marginal problem on finite sets 235 V. L. LEVIN / Topics in the duality for mass transfer problems 243 C. S. SMITH, M. KNOTT / Generalising monotonicity 253 L. RUSCHENDORF, 1. UCKELMANN / On optimal multivariate cou-
plings 261 L. UCKELMANN / Optimal couplings between one-dimensional distri-
butions 275 D. RAMACHANDRAN, 1. RUSCHENDORF / Duality theorems for
assignments with upper bounds 283 J. H. B. KEMPERMAN / Bounding the moments of an order statistics
if each k-tuple is independent 291 SUBJECT INDEX 305
PREFACE
The last decade has seen a remarkable development of the "Marginal and Moment Problems" as a research area in Probability and Statistics. Its attractiveness stemmed from its lasting ability to provide a researcher with difficult theoretical problems that have direct consequences for appli cations outside of mathematics. The relevant research aims centered mainly along the following lines that very frequently met each other to provide sur prizing and useful results :
-To construct a probability distribution (to prove its existence, at least) with a given support and with some additional inner stochastic property defined typically either by moments or by marginal distributions.
-To study the geometrical and topological structure of the set of prob ability distributions generated by such a property mostly with the aim to propose a procedure that would result in a stochastic model with some optimal properties within the set of probability distributions.
These research aims characterize also, though only very generally, the scientific program of the 1996 conference "Distributions with given marginals and moment problems" held at the beginning of September in Prague, Czech Republic, to perpetuate the tradition and achievements of the closely related 1990 Roma symposium "On Frechet Classes" 1 and 1993 Seattle" AMS Summer Conference on Marginal Problem".
A careful selection of the contributions presented to the 1996 conference is published in these proceedings. It has been an aim of the editors not only to reflect the recent developments in the field, but also, with an indispens able assistance of numerous referees, to publish new and important results. These are the reasons that make the editors believe that the book will prove itself useful even for the probabilists and statisticians working outside its specialized domain and will surely stimulate the research in the "Moments & Marginals" area itself.
1 For the proceedings see "Advances in Probability Distributions with Given Ma.rginals" . Kluwer Academic Publishers, Dodrecht 1991.
vii
viii
Roughly speaking, the contributions of the proceedings belong to one of the following topics centered around the invited papers written be carefully selected scholars :
- Moment Problems and Applications. G.A.Anastassiou applies Geometric Moment Theory to a statistical problem over observations sub jected to one moment condition. L.B.Klebanov and S. T.Rachev introduce Computer Tomography and Quantum Mechanics as research fields where the moment and marginals technologies may be applied to get new intrigu ing results - a solution of the computer tomography paradox, for example. J.H.B.Kemperman shows how to calculate lower and upper bounds of the moments of an order statistics if each k-tuple in the sample is formed be independent random variables with a known probability distribution.
- Marginal Problems and Stochastic Order. M.Scarsini and M.Sha ked examine in their paper non monotone dependence bivariate stochastic models in the framework of a Frechet class. By nonmonotone dependence is meant the tendency of a bivariate distribution to concentrate its probability mass around the graph of a (not necessarily monotone) function.
- Copulas. This well established topic with a remarkable history in the field is represented by "On Approximation of Copulas" by X.Li, P.Mi kusinski, H.Sherwood, and M.D. Taylor. The authors observe that uniform convergence of copulas is not completely satisfactory when trying to verify some of their properties that could be derived first on a suitable class of "simple" copulas and then extended to all copulas by a limiting process. Two different ways of approximating copulas are proposed (Checkerboard and Bernstein approximations) to meet the needs mentioned above.
- Marginal and Moment Problems - Measure Theory Approach. G.Letac comes back to a problem with the long history namely that of find ing extremal measures in the convex set of probability measures with a fixed pair of marginals. The problem is attacked, this time, in the framework of Lanchester R 2-probabilities. H.G.Kellerer treats the problem of represent ing a Markov kernel that connects a pair of ordered spaces by an order preserving random map. The problem is a natural counterpart to the pre vious studies on Markov kernels represented by continuous random maps.
- Stochastic Programming and Artificial Intelligence. J.Dupa cova discusses in her paper different methods for bounding the optimal values of stochastic programs, in particular for recourse problems. R.Jirou sek studies the problem of constructing a distribution with given marginals as an inherent part of artificial intelligence approaches to applied problems. In this context, only the constructions that can be handled efficiently by computer programs are acceptable.
- Optimizations in Marginal Problem The paper by L.Levin con cerns the duality theory for nontopological versions of mass transport prob-
ix
lem with the aim to extend the classical Monge-Kantorovich results. L.Rii schendorf and 1. Uckelmann treat optimal multivariate problems, prove the existence of optimal Monge solutions in this context and propose an ex plicit construction method for the optimal transportation in the case that one mass distribution is discrete. Finally, D.Ramachandran and L.Riischen dorfformulate a version of the nonatomic assignment .problem with upper admissible assignments in the form of a dominating measure and obtain a general duality theorem for this model.
The invited papers, the achievements of which have been reviewed above are accompanied by 25 contributed papers that, according to the editors' opinion, increase significantly the scientific value of the proceedings. Cordial thanks are due to all authors and referees.
Prague, January 1997
ACKNOWLEDGEMENTS
The editors desire in the first place to acknowledge that it would never have been possible to organize the 1996 Prague Conference "Distributions with Given Marginals and Moment Problems" and to prepare its Proceed ings for the publication if it had not been for the research grant they have received from the Grant Agency of the Czech Republic in 1994 under the contract 201/94/0471. The grant not only encouraged their own sci entific work but also provided an invaluable help in keeping and promoting necessary international contacts.
The editors are no less indebted to Faculty of Mathematics and Physics of Charles University in Prague, Faculty of Mechanical Engineering of Czech Technical University in Prague, Institute of Information Theory and Automation in Academy of Sciences of Czech Republic and to Technical Faculty of Czech Agricultural University in Prague which institutions spon sored generously the Conference activities.
Among many individuals to whom the organizers of the Conference and the editors of its Proceedings owe for advice, help and encouragement are firstly the members of the Conference scientific and organizing committees: L. Riischendorf (Freiburg i. Br.), H. Sherwood (Orlando), S. T. Rachev (Santa Barbara), J. Dupacova, V. Slavik, M. Studeny, P. Lachout, D. Hlu binka, T. Visek.
D. Hlubinka and T. Visek also have the editor's grateful thanks for preparing the manuscript technically. Last but not least cordial thanks are due to H. Jandova for the skill and patience with the editors when mastering their extensive correspondence and paperwork.
Viktor Benes and Josef Stepan
x
OFF OBSERVATION IN THE PRESENCE OF A SINGLE MOMENT
CONDITION
GEORGE A. ANASTASSIOU Department of Mathematical Sciences The University of Memphis Memphis, TN 38152 U.S.A.
Abstract. The moment problem of finding the maximum and minimum expected values of the averages of nonnegative random variables over various sets of observations subject to one simple moment condition is encountered. The solution is given by means of geometric moment theory (see [3]).
1. Background
Here [.j and r'l stand for the integral part (floor) and ceiling of the number, respectively.
We consider probability measures p, on A := [0, aj, a > ° or [0, +00). We would like to find
Ua := sup f ([t] + rtl) p,(dt) p. JA 2
and
subject to
L tT J.L( dt) = dT , r > 0,
where d > ° (here the subscript Q in Ua, La stands for average).
V. Bend and J. Stepan (ew.). Distributions with given Marginals and Momenl Problems. 1-13. @ 1997 Kluwer Academic Publishers.
(1)
(2)
(3)
2 GEORGE A. ANASTASSIOU
In this article, the underlined probability space is assumed to be nonatomic and thus the space of laws of nonnegative random variables coincides with the space of all Borel probability measures on R+ (see [4], pp. 17-19).
The solution of the above moment problems are of importance because they permit comparisons to the optimal expected values of the averages of nonnegative random variables, over various sets of observations, subject to one simple and flexible moment condition.
Similar problems have been solved earlier in [1], [2]. Here again the solutions corne by the use of tools from the Kemperman geometric moment theory, see [3]. Here of interest will be to find
and
subject to the one moment condition (3). Then, obviously,
(4)
(5)
(6)
So we use the method of optimal distance in order to find U, L as follows: Call
M :== cOnvtEAW, ([tl + ftl )). (7)
When A = [0, a] by Lemma 2 of [3J we get that ° < d ~ a, in order to find finite U and L. Then U = sup{z: (dT,z) E M} and L = inf{z : (dT,z) E M}. Thus U is the largest distance between (dT,O) and (dT,z) EM. Also L is the smallest distance between (dT , 0) and (dT , z) EM. SO we are dealing with two-dimensional moment problems over all different variations of A :=
[0, a], a > 0, or A := [0, +00) and T ~ 1 or ° < T < 1. We intend elsewhere to solve the corresponding three-dimensional mo
ment problems, where we include one more moment condition satisfied by JL.
Description of the graph (XT, ([xJ + f xl)), T > 0, a ~ +00. This is a stairway made out of the following steps and points, the next step is two units higher than the previous one with a point included in between, namely we have: the lowest part of the graph is Po = (0,0), then open interval {PI = (0,1), Qo = (1, I)}, then included point (1,2), open interval {P2 = (1,3), QI = (2T,3)}, included point (2T,4), open interval {P3 = (2T,5), Q2 = (3T,5)}, included point (3T,6), ... , {open intervals {PHI := (kT,2k + 1), Qk := ((k+lY, 2k+ I)}, included points (kT, 2k); k = 0,1,2,3, ... }, ... , keep going like this if a = +00, or if ° < a ¢ N then the last top part, segment
OPTIMAL BOUNDS ON THE AVERAGE 3
ofthe graph, will be the open closed interval {([ar, 2[aJ + 1), (aT, 2[aJ + I)}, or if a E N then the last top part, included point, will be (aT, 2a). Here Pk+2 = ((k + 1Y, 2k + 3). Then
which is decreasing in k if r ~ 1, and increasing in k if ° < r < 1. The closed convex hull of the above graph M changes according to values of r and a.
Here X stands for a random variable and E for the expected value.
2. Results
Our first result follows
Theorem 1. Let d > 0, r ~ 1 and 0< a < +00. Consider
U := sup{E([XJ + rXl): ° ~ X ~ a a.s., EXT = dT }. (8)
1) If k ~ d ~ k + 1, k E {O, 1,2, ... , [ aJ- I}, a ¢ N, then
2dT + (2k + 1)(k + 1y - (2k + 3)kT U= .
(k + ly - kT (9)
U = 2[a] + 1. (10)
3) Let A = [0, +00), (i.e., a = +00) and k ~ d ~ k + 1, k = 0,1,2,3, ... , then U is given by (9).
4) Let a E N - {I, 2} and k ~ d ~ k + 1, where k E {O, 1,2, ... , a - 2}, then U is given by (9).
5) Let a E N and a-I ~ d ~ a, then
U = dT + aT(2a - 1) - (a - lY2a. aT - (a - ly (11)
Proof. We are working on the (XT, y) plane. Here the line through the points Pk+I, Pk+2 is given by
2xT + (2k + 1)(k + It - (2k + 3)P y = (k + 1 Y - kT •
(12)
In the case of a E N let Pa := ((a - l)r, 2a - 1). Then the line through Pa
and (ar , 2a) is given by
(13)
The above lines describe the closed convex hull !VI in the various cases mentioned in the statement of the theorem. Then we apply the method of optimal distance described earlier in §1. 0
Its counterpart comes next.
Theorem 2. Let d > 0, 0 < r < 1 and 0 < a < +00. Consider
U := sup{E([X] + rXl):O::; X::; a a.s.,EXT = dT }. (8)
1) Let 0 < a ¢ N and [a] ::; d::; a, then
U = 2[a] + 1. (14)
2) Let 0 < a ¢ Nand 0 < d::; [a], then
2[a] U = [a]T ~ + 1. (15)
2*) Let a = 1 and 0 < d :::; 1, then U = dr + 1. 3) Let a E N - {I}, 0 < d :::; a and 0 < r ::; !, then
( 2a - 1) U= -;r- ~+l. (16)
4) Let a E N - {I}, 0 < d::; a-I, ! < r < 1 and a ~ (2Tt ,then (2r) -r-l
U = 2(a _1)I-T~ + 1. (17)
1 (2Tt h 5) Let a E N - {I}, a-I::; d::; a, '2 < r < 1 and a ~ ,t en (2T) -r-l
(18)
U = +00. (19)
Proof. We apply again the method of optimal distance. From the graph of (XT, ([X] + r xl)) we have that the slope (PHI Pk+2) = (k+1)r-kr is increasing in k (see §1).
Let 0 < a ¢ N, then the upper envelope of the closed convex hull if is formed by the line segments {PI = (0,1), P[aJ+I := ([aY, 2[a] + I)} and {p[a)+l> (aT, 2[a] + I)}. Then case (1) is clear ..
The line (PI p[a)+1) in the (x r , y) plane is given by
y = 2[a]xT + 1. [aV
Therefore case (2) follows immediately. Case (6) is obvious by the nature of M when 0 < r < 1. N ext let a E N - {I} and 0 < r ~ !. I.e.,
( ~ )l-T 2r~1< a-I ,forany~E(a-1,a).
(20)
That is 2(a - l)I-T < re!-i' any ~ E (a - 1, a). Hence by the mean value theorem we obtain
2 a-I I-T < 1 ( ) aT - (a - 1Y (21)
Therefore slope(mo) < slope(mt), where mo is the line through the points PI = (0,1), Pa = ((a - 1Y, 2a - 1), and ml is the line through the points Pa , (aT, 2a) . In that case the upper envelope of if is the line segment with endpoints PI = (0,1) and (aT, 2a) . This has equation in the (XT, y) plane
( 2a - 1) Y = --;;;- xT + 1. (22)
The last implies case (3). Finally let a EN - {I} and! < r < 1, along with
1 (2r)r=T
Hence
(_a_) ~ (2r) l':r . a-I
( ) I-T I.e., a~I ~ 2r and e-T < aI- T ~ (a - I)I-T2r, any ~ E (a - 1, a).
Thqs e-T ~ 2r(a _1)I-T, and T~!-l ~ 2(a - l)l-T, any ~ E (a - 1, a). And by the mean value theorem we obtain that
1 slope(mI)= ( ) ~slope(mo)=2(a-l)I-T,
aT - a-I T
where again ml is the line through Pa = ((a-lY, 2a-l) and (aT, 2a), while mo is the line through PI = (0,1) and Pa. Therefore the upper envelope of M is made out of the line segments PI Pa and Pa (aT, 2a). The line mo in the (x T , y) plane has equation
Now case (4) is obvious. The line ml in the (XT, y) plane has equation
xT + 2a( aT - (a - 1 Y) - aT y=
aT - (a - lY A t last case (5) is clear.
(23)
(24)
o Next we determine L in different cases. We start with the less compli
cated case where 0 < r < 1.
Theorem 3. Let d > 0, 0 < r < 1 and 0 < a < +00. Consider
L:= inf{E([X] + rXl):O ~ X ~ a a.s., EXT = dT }. (25)
1) Let 1 < a ¢ Nand 0 < d ~ 1, then
(26)
1 *) Let a < 1, 0 < d ~ a, then L = a -T dT •
2) Let 2 < a ¢ Nand k + 1 ~ d ~ k + 2 for k E {O, 1, ... , [a] - 2}, then
L = 2dT + (2k + l)(k + 2Y - (2k + 3)(k + lY. (27) (k+2Y-(k+1Y
3) Let 1 < a ¢ N and [a] ~ d ~ a, then
L _ 2dT + aT (2[a] - 1) - [aY(2[aJ + 1) - ~_~]T . (28)
4) Let a E Nand 0 < d ~ 1, then
L = dT. (29)
5) Let a E N - {I} and k + 1 :::; d :::; k + 2, where k E {O, 1, ... , a - 2}, then
L is given by (27).
6) Let A:= [0,+00) (i.e., a = +00) and 0 < d:::; 1, then
7) Let A := [0, +00) and k + 1 :::; d:::; k + 2, where k E {O, 1,2, ... }, then L is given by (27).
(30)
Proof. The lower-envelope of the closed convex hull M is formed by the points Po:= (0,0), {Qk := ((k + lY, 2k + 1), k = 0,1,2,3, ... }, Q[al-l := ([ar,2[a]-I), F:= (ar,2[a] + 1). Here
which is increasing in k since 0 < r < 1, and slope(PoQo) = 1. Notice that slope(QoQl) = 2r:l > 1. When ° < a ¢ N we observe that
Therefore the lower envelope of M is made out of the line segments PoQo, QOQb QIQ2' .. ·' QkQk+b ... , Q[a]-2Q[a]-1 and Q[al-I F . In the cases of a E N or a = +00 we have a very similar situation. Cases (1), (4) and (6) are obvious.
The line (QkQk+d has equation in the (xr,y) plane
2xr + (2k + 1)(k + 2Y - (2k + 3)(k + lY (31) y= (k+2Y-(k+l)r .
Now the cases (2), (5) and (7) follow easily by the principle of optimal distance.
Finally the line (Q[al-lF) in the (xr,y) plane has equation
2xr + ar(2[a]- 1) - [an2[a] + 1) y = ar - [a]r . (32)
Thus, case (3) now is clear by applying again the principle of optimal (min imum) distance. 0
The more complicated case of L for r ~ 1 follows in several steps. Here the lower envelope of the closed convex hull M is formed again by the points Po = (0,0), Qo = (1,1), ... ,Qk = ((k+ ly,2k+ 1),.·.,Q[al-l = ([a]r,2[a] -1), F = (ar,2[a] + 1) or F* = (ar,2a) if a E N. In the case of
A = [0, +00), the lower envelope of if is formed just by Po, Q k; all k E Z+. Observe that the
is decreasing in k since r 2: 1.
Theorem 4. Let A = [0, +00), (i.e., a = +00) r 2: 1 and d > O. Consider
L:= inf{E([X] + rXl):X 2: 0 a.s., EXT = dT}. (33)
We meet the cases: 1) If r > 1 we get that L = O. 2) If r = 1 and 0 < d ~ 1, then L = d. 3) If r = 1 and 1 ~ d < +00, then L = 2d - 1.
Proof. We apply again the principle of optimal (minimum) distance. When r > 1 the lower envelope of if is the x T -axis. Hence case (1) is
clear. When r = 1, then slope(QkQk+d = 2 any k E Z+ and slope(PoQo) = 1.
Therefore the lower envelope of if here is made by the line segment (PoQo) with equation y = x, and the line (QoQ 00) with equation y = 2x - 1. Thus cases (2) and (3) are established. 0
Theorem 5. Let a E N - {l}, d> 0 and r 2: 1. Consider L as in (25).
1) If 2~~:::~) 2: 1 and 0 < d ~ 1, then
2) If 2!~:::~) 2: 1 and 1 ~ d ~ a, then
L= 2(a-l)dT+(aT -2a+l). aT -1
3) If 2i~=~) ~ 1 and 0 < d ~ a, then
(34)
(35)
(36)
Proof. Here mo := line(PoQo) has equation y = x T and slope(mo) = 1. Also ml := line(QoQa-l), where Qa-l = (aT, 2a - 1), has equation
2( a-I )XT + (aT - 2a + 1) y = T 1 ' a -
(37)
. h I ( ) 2(a-l) WIt s ope ml = ar-l .
Notice also that slope(QkQk+d is decreasing in k. If 2i~=~) ? 1, then
the lower envelope of M is made out of the line segments PoQo and QOQa-l. Hence the cases (1) and (2) are now clear by application of the principle of minimum distance.
If 2~~=~) ~ 1, then the lower envelope of M is the line segment (POQa-l) with associated line equation
( 2a - 1) T y= -- x. aT
Thus case (3) is established similarly.
Theorem 6. Let 2 < a ¢ N, n := [a], d> 0 and r ? 1. Assume that
1 < 2(n - 1) < 2
Consider L as in (25). 1) If 0 < d ~ 1, then
2) If 1 ~ d ~ [a], then
- nT - 1 - aT - nT
L = 2([a]- l)dT + ([a]T - 2[a] + 1). [ay - 1
3) If [a] ~ d ~ a, then
L = 2dT + aT(2[a] - 1) - [a]T(2[a] + 1). aT - [a]T
(38)
o
(39)
(40)
(41)
(42)
Proof. The lower envelope of M is made out of the line segments: {Po = (0,0), Qo = (1, I)} with equation y = XT, {Qo = (1,1), Q[a]-l = ([ar, 2[a] I)} with equation
2([a]- l)xT + ([a]T - 2[a] + 1) Y = [a]T - 1 '
and {Q[a]-l, F = (aT, 2[a] + 1)} with equation
2xT + aT(2[a] - 1) - [aY(2[a] + 1) Y = aT - [a]T .
Then cases (1), (2), (3) follow immediately by application of the principle of minimum distance as before. 0
Theorem 7. Let 2 < a rJ. N, n := [aJ, d > 0 and r 2: 1. Assume that
2(n - 1) 2 1> > . - nT - 1 - aT - nT (43)
Consider L as in (25). If 0 < d::; a, then
(44)
Proof. Here the lower envelope of M is made out of the line segment {Po = (0,0), F = (aT, 2[a] + I)} with equation
y = (2[al T + 1) XT,
Etc,
Theorem 8. Let 2 < a rJ. N, n := [aJ, d> 0 and r 2: 1. Assume that
and
Consider L as in (25). 1) If 0 < d $ [aJ, then
2) If [a] $ d $ a, then
2(n - 1) < 2 < 1 nT - 1 - aT - nT -
2n - 1 2 --<--nT aT - nT
L = (2[a] - 1) dT, [a]T
L _ 2dT + aT (2[a]- 1) - [ar(2[a] + 1) - aT - [a]T '
o
(45)
(46)
( 47)
(48)
Proof. Here the lower envelope of M is given by the line segments: {Po = (0,0), Q[a]-l = ([ar, 2[a]- I)} with equation
_ (2[a]- 1) T
Y - [a]T x
and {Q[a]-l, F = (aT, 2[a] + I)} with equation
2xT + aT (2[a]- 1) - [a]T(2[a] + 1) Y = aT - [a]T ' (49)
o
Theorem 9. Let 2 < a ¢ N, n := [aJ, d > 0 and r 2: 1. Assume that
2(n - 1) < 1 < 2 nT - 1 - - aT - nT (50)
Consider L as in (25). Then L is given exactly as in cases (1) and (2) of Theorem 8.
Proof. The same as in Theorem 8.
Theorem 10. Let 2 < a ¢ N, n := [aJ, d > 0 and r 2: 1. Assume that
and
Consider L as in (25). 1) If 0 < d ~ 1, then
2) If 1 ~ d ~ a, then
2 < 1 < 2(n - 1) aT - nT - - nT - 1
2n -->1. aT - 1
L = 2[a]dT + (aT - 2[a]- 1). aT - 1
o
(51)
(52)
(53)
(54)
Proof. Here the lower envelope of M is made out of the following line segments: {Po = (0,0), Qo = (1, I)} with equation y = xr, and {Qo, F = (a T ,2[a] + I)} with equation
2[a]xT + (aT - 2[a]- 1) y = aT - 1 . (55)
Etc. o
Theorem 11. Let 2 < a ¢ N, n := [aJ, d> 0 and r 2: 1. Assume that
2 2(n- 1) 1< < .
- aT - nT - nT - 1 (56)
Consider L as in (25). Then L is given exactly as in cases (1) and (2) of Theorem 10.
Proof. Notice that the lower envelope of M is the same as in Theorem 10, especially here a;~l > 1. Etc. 0
Theorem 12. Let 2 < a ~ N, n := [a], d> 0 and r ~ 1. Assume that
2(n -1) < 2 < 1 nT - 1 - aT - nT - (57)
and 2n -1 2 -->---nT - aT - nT (58)
Consider L as in (25). If 0 < d ~ a, then
(59)
Proof. Here the lower envelope of Sf is the line segment PoF with equation
(60)
Etc. o
Theorem 13. Let 2 < a ~ N, n := [a], d > 0 and r ~ 1. Assume that
2 < 1 < 2(n - 1) aT - nT - - nT - 1
(61)
(63)
Proof. Same as in Theorem 12. o
Proposition 1. Let a = 1, r ~ 1 and 0 < d ~ 1. Then L = dT.
Proposition 2. Let 0 < a < 1, r ~ 1 and 0 < d ~ a. Then L = a-TdT.
Proposition 3. Let a = 2, r ~ ~ and 0 < d ~ 2. Then
Proposition 4. Let a = 2, 1 ~ r ~ ~~. 1) If 0 < d ~ 1, then L = dT.
2dr 2) If 1 ~ d ~ 2, then L = 2r-l'
Proposition 5. Let 1 < a < 2, r ~ 1 and 2(ar - 1)-1 > 1. 1) If 0 < d ~ 1, then L = dr. 2) 111 < d < a then L = 2dr tar -3.
- - , ar-l
Proposition 6. Let 1 < a < 2, r ~ 1 and 2( ar - 1)-1 ~ 1. IfO < d ~ a, then L = 3a-r dr •
Open Problem. In (1) and (2) replace (ltl~[tl) by (.x[t] + (1- .x)ftl),
where.x E (0,1), .x -:f ~. Then find the general Ua, La .
Acknowledgement. The author wants to thank the referee for making interesting comments.
References
[1] G.A. ANASTASSIOU, Moments in probability and approximation theory, Longman Sci. & Tech., Harlow, UK, 1993.
[2] G.A. ANASTASSIOU and S.T. RACHEV, Moment problems and their applications to characterization of stochastic processes, queueing theory, and rounding problem, in Proc. 6th S.E.A. Meeting on Approximation Theory, pp. 1-77, Marcel Dekker, New York, 1992.
[3] J .H.B. KEMPERMAN, The general moment problem, a geometric approach, Ann. Math. Statist. 39(1968), 93-122.
[4] S.T. RACHEV, Probability metrics and the stability of stochastic models, John Wiley & Sons, New York, 1991.
THE COMPLETE SOLUTION OF A ROUNDING PROBLEM
UNDER TWO MOMENT CONDITIONS
"
TOMASZ RYCHLIK Institute of Mathematics, Polish Academy of Sciences, Chopina 12, 87100 Torun, Poland
Abstract. We complete a partially known solution of the problem of deter mining the extreme expectations of arbitrarily rounded nonnegative (pos sibly bounded) random variables with given two moments: the expectation and another moment of a positive order.
1. STATING THE PROBLEM AND INTRODUCTION
We consider the family of nonnegative random variables with a given, pos sibly infinite range and two fixed moments
o ~ X ~ a, a.s., 0 < a ~ +00, EX = mI,
O<r=J.l.
Our problem consists in determining the accurate bounds for the expec tation of c-rounded variable E[X]e, 0 ~ c ~ 1, where the integer valued c-rounding function [']e is defined by
[x]e = inf {k E Z: x ~ k + c}, 0 ~ c < 1,
and [x h = inf {k E Z: x < k + 1}.
The cases c = 0 and c = 1 describe the popular Adams and Jefferson rules of rounding, or the ceiling and floor of a number, respectively. We shall write
L(mI' mr )
inf E[X]e, sup E[X]e,
V. Benes and J. ~tipcln (eels.), Distributions with given Margina/s and Moment Problems, 15-20. © 1997 Kluwer Academic Publishers.
16 TOMASZ RYCHLIK
ignoring in notation the dependence of extremes on the other parameters: a, rand c.
For a number of subcases, the problem was solved in Anastassiou and Rachev (1992) and Anastassiou (1993, Chap. 4) by means of a geometric approach to moment problems due to Kemperman (1968). The crucial tool of the geometric moment theory, proved independently by Richter (1957) and Rogosinsky (1958), is the fact that for every probabilistic measure with given number of moment conditions, there exists a probability distribution with the same moments as the original one which is supported on a finite number of points, exceeding the number of conditions by no more than one.
To adopt the theory for our problem, we introduce some useful notation. Given 0 < r 'II and 0 ~ c ~ 1, put t = (t,tT) E R2 and T = (t,tT,[t]e) E R 3 , 0 ~ t ~ a. We shall further adhere to the convention that points of the plane and space are written in bold lower case and capital letters, respectively (e.g., 0 = (0,0) and 0 = (0,0,0)). We denote S]S2 = {t E R2 : SI ~ t ~ S2}. Also, let 8182 and 6818283 stand for the line segment and triangle, respectively, determined by their extreme points.
The problem is well posed iff m = (ml' mT ) E cooo, the region between the curve tT and the line aT-It, 0 ~ t ~ a. The solution consists in deter mining the endpoints ML = (m, L(m)) and Mu = (m, U(m)) of the line segment arising by intersecting the moment space co{T E R3: 0 ~ t ~ a} by the vertical line t = m. Since t t-t t is continuous and t t-t [t]e, 0 ~ c < 1, is lower semi continuous , and t t-t [tlt is upper semicontinuous, inf E[X]e, o ~ c < 1, and sup E[Xh are attained (see Kemperman (1968, Remark on p. 99)). To make possible determining the unattainable extremes, we should add Lk = (tk, k) = (k+c, (k+ct, k) and Uk = (tk' k+ 1), 0 ~ k < [ale = n, say, to the moment space (see Kemperman (1968, Theorem 6)).
2. THE SOLUTION
We proceed to review the results of Anastassiou and Rachev (1992) and then to present new ones. We have
L(m)
(1) (2)
if m E CO{tk: 0 ~ k < n}. Note that these trivial bounds, implied by X ~ [X]e + C ~ X + 1, are the best ones here. For c < 1, (1) is attained by any distribution on k + c with probabilities o.k, 0 ~ k < n, satisfying L o.ktk = m. Bound (2) is merely approximated by the sequences supported on k + c + €kj, 0 ~ k < n, €kj ! 0 as j -+ 00, and obeying the moment constraints. (For the Jefferson rule, it is the supremum that
ROUNDING UNDER TWO MOMENT CONDITIONS 17
is attained). If m E coot;;, then L(m) = 0, being attained by any pair o ~ SI,S2 ~ c such that m E 81,82.
Suppose that a < +00 and m to the tetragon with the vertices 0, to, tn- l and a. If A = (a, n) lies above the plane containing 0, Lo and Ln-I, then
and
alO + a2LO + a3Ln-l, for mE 6.ototn_I, alO + a2Ln-l + a3A, for m E 6.otn- l a,
ML - { alO + a2 LO + a3A , for mE 6.otoa, - alLo + a2Ln-l + a3A, for mE 6.tOtn_l a,
(3)
(4)
otherwise. In each formula of (3) and (4), the vertices of triangles define the support of the (limiting) extreme distribution, the first and second equa tions enable us to calculate the coefficients of the convex combination, and the last one provides L( m). To derive U( m), it suffices to replace Lo, Ln- l and ML in (3)-(4) and the preceding condition by Uo, Un- l and Mu, re spectively. The explicit expressions for the extremes are complicated and we do not present them here (see Anastassiou (1993, pp. 91-100)). More over, for m E cot;:;a, we have U(m) = n, attained by a combination of some SI, S2 E [n, a].
If a = +00 and r < 1, the for all m from the halfstrip 0 ~ mr ~ cr ,
ml ~ c1-rmn we have L(m) = ml - c1-rmr and U(m) = c-rmr • For a = +00, and r > 1, and m such that 0 ~ ml ~ c, mr ~ cr-lml, it follows that L(m) = 0 and U(m) = C-lml. Except the cases that the moments belong to the joint border of the two regions, the extremes are unattainable and approached by measures with a support point running to infinity and the respective probability vanishing at an appropriate rate.
Neither of extremes is known, when m E cott:;tk, 1 ~ k < n. Also, there are lacking the upper one for m E coot;;, and the lower one for bounded random variables with m E cot;:; a. Such pairs of moments come from variables highly concentrated around the expectation.
Proposition 1. Suppose that m E cotk-:;tk for some 1 ~ k < n. Let VL, vu E [k - 1 + c, k + c] and a, f3 E [0,1] be uniquely defined by the equations
Then
18 TOMASZ RYCHLIK
and the limiting distributions of the sequences approximating (7) and (8) are the two-point distributions:
respectively.
Pr(X = k + c) = 1- Pr(X = vu) = {3,
Proof We examine the infimum problem, only. The other one may be handled in much the same way and we leave it to the reader to adapt the following reasoning for it. Using arguments of the geometric moment theory, we can confine ourselves to the random variables distributed on three point, say v, wand z, at most. Due to the two moment conditions, the distributions are determined by the support points. The idea of proof is to provide other values such that the respective moment point M = (ml' mT , m(c)) E R 3 ,
m(c) = E[X]c, has a smaller third coordinate than the assumed distribution for all possible positions of v, w, z. Consecutive improvements lead to the assertion. It is evident that some numbers among v, wand z must belong to [k - 1 + c, k + c], since otherwise m ¢ cotk~tk' Because there are random variables with assumed moments ml and mT , and m(c) < k (see the statement of the Proposition), we can conclude that the candidates for the infimum must have at least one support point less than or equal to k - 1 + c.
We first consider the case 0 5 v 5 k - 1 + c < w < z 5 k + c. The first step consists in replacing wand z by a single point. Let M = (m, m(c)) E 6.VWZ and V' be the element of the line segment WZ such
that M E VV'. Let v" denote the point at which t';;tk and the half line '0'0' ...... , beginning at v and passing through v', cross each other. Observe that the halfline VV" ...... runs beneath VV' ...... , because they reach level k at v" and v', respectively. Therefore, for M' = (m,m(c)) E VV", we have
m(c) 5 m(c) ' If v = k - 1 + c, this is the final claim. Otherwise we proceed to the second step, in which we present a way of
eliminating points lying outside of [k-l +c, k+c] . Suppose that v < k-l + c < z 5 k + c and M En. Define v' = vz n tk-ltk, V' = (v',v(c)) E VZ,
and V" = (v',v(~)) ELk-ILk. By (1) , v(~) = L(v') = v~ - c 5 v(c) so that
the whole V"Z is below V'Z . In particular, for M' = (m,m(c)) E V"Z yields m(c) 5 m(c)' On the other hand, M' E 6.Lk-lZLk, and applying the arguments of the first step, we can reduce m(c) by taking a single point instead of Z and Lk .
It remains to treat the case of two support points, say wand z, situated apart from (k - 1 + c, k + c]. As in the second step of the previous case, we replace both Wand Z by W', Z' ELk-ILk so that 6. VW' Z' with
ROUNDING UNDER TWO MOMENT CONDITIONS 19
the modified M' lie under L VW Z and M, respectively. Since M' may be represented as a combination of Lk-b Z and Lk, it suffices to take the first step again to derive the desired conclusion. 0
The method of proof carries over to the remaining open problems.
Proposition 2. If m E co~, then U(m) = {3 for some 0 ~ f3 ~ 1 and o ~ Vu ~ c satisfying
(1 - (3)vu + (3to = m. (9)
Proposition 3. If a < +00 and m E cot~a, then L(m) = n - a for o ~ a ~ 1 and n - 1 + c ~ VL ~ a such that
(1 - a)vL + atn - I = m. (10)
In contrast with the previously known solutions for which the points of the lower and upper envelopes of the moment space compose pieces of several plane~we here obtain curved surfaces. E.g., the part of the ~ce for m E cotk-Itk consists of two cones with the common base COUk-lLk and vertices Lk-I and Uk. This is the implication of the fact that each extreme distribution has two support points. One is in common for various moment conditions and yields either Lk-I (the infimum case) or Uk in the moment space. The moments of the other vary along the arc Uk~k.
Only for a few specific r we can solve equations (5), (6), (9) and (10) analytically and derive explicit expressions for Land U in Propositions 1-3. For r = 2 and r = ! we have
L(m)
U(m)
and
k - 2" 2 ' (m! - vlk + 1- c)2 + ml - m 1
2 2"
U(m)
respectively. The bounds hold for 1 ~ k < n if m E cotk~tk. The upper ones are also true with k = 0 as m E co04;, and the lower ones apply to k = nand mE cot:=;a (for finite a). They become more meaningful when
20 TOMASZ RYCHLIK
we realize that m2 - m~ =VarX and ml - m~ = Van/X. In the case r = 3, 2"
we obtain (7) and (8) with a and f3 being expressed by huge formulas as the greater and smaller roots of the following quadratic equations:
(m3-a~k-l+c)3
(1 - f3)2[ml - a(k + c)],
respectively. The ranges of application coincide with those given above. Similar formulas can be concluded for r = ~ .
We finally point out that simply putting c = 0 and 1 we derive the extreme expectations for the Adams and Jefferson rounding proportions, respectively.
References
Anastassiou, G .A. and Ra.chev, S.T . (1992) Moment problems and their applications to characterization of stochastic processes, queueing theory, and rounding problem. In: Proc. 6th S.E.A . Meeting "Approximation Theory". Marcel Dekker, New York, pp. 1-77.
Kemperman, J .H.B. (1968) The genera.! moment problem, a geometric approach, Ann. Math. Statist., Vol. no. 39, pp. 93-122.
Richter, H. (1957) Para.meterfreie Abschatzung und Rea.lisierung von Erwartungswer ten, Blatter der Deutschen Gesellschaft for Versicherungmathematik, Vol. no. 3, pp. 147-161.
Rogosinsky, W.W. (1958) Moments of non-negative mass, Proc. Roy. Soc. London Ser. A, Vol. no. 245, pp. 1-27.
METHODS OF REALIZATION OF MOMENT PROBLEMS
WITH ENTROPY MAXIMIZATION
VALERIE GIRARDIN UFR Sciences, Universite de Caen, BP5186, 14032 Caen, Statistique, Batiment 425, UPS, 91405 Orsay Cedex, France [email protected]·fr
Let K be a finite subset of 71/ including 0 = (0, ... ,0). Let c) = (¢>k)kEK be a family of measurable and linearly independent complex or real func tions defined over a compact I C lRr , with ¢>o == 1. Let p be a reference measure over I.
We construct here h maximizing an entropy functional in both the fol lowing problems
(1)
where C = (Ck)kEK is positive in the sense of Krein & Nudelman [5] (if (ak)kEK is such that LkEK ak¢>k ~ 0 over I then Lk akCk ~ 0), and
(2)
where D = (dkd(k,I)EK2 is a positive definite matrix. We call PE the subspace of polynomials of L;(I) spanned by (¢>k)kEE.
Problem (1) is a generalized moment problem. Relation (2) defines a d-scalar product over PK by
Hence Problem (2) is a problem of realization of this d-scalar product. It may also be considered as a moment problem for the family (¢>k¢>d(k,I)EK2.
This has a probabilistic interpretation. Let D be the autocovariance ma trix of a purely nondeterministic multidimensional process X = (Xk)kE,Zr.
21
V. Benes and 1. Stipdn (eds.), Distributions with given Marginals and Moment Problems, 21-26. © 1997 Kluwer Academic Publishers.
22 VALERIE GIRARDIN
It admits a stationary realization if there exists a nonnegative bounded measure J.L over the torus nr such that
The process admits then the representation
where Z( >.) is a process with orthogonal increments and energy J.L. If J.L is absolutely continuous with respect to the Lebesgue measure A over nr , its density h is the spectral density of the process. Note that the coefficients dkl of D are functions of k - I only so that Problem (2) becomes here a problem (1) for a set of indices of the form K - K.
By extension, see Priestley [7], if D admits a representation
dkl = 14>k(>')<M>')h(>')dP(>'), (k,l) E yzr x yzr
for a given family ~, then X admits the spectral representation
(3)
The functions 4>k may have a precise physical meaning, real exponentials for dampings, algebraic monomials for generalized power spectra, product of a function by complex exponentials for the output process of a linear system ...
So, (2) is the problem of representing the covariance matrix of (Xk)kEK by a measure of density hand (1) is a moment problem for the process X.
We are interested here in solutions maximizing an entropy functional. Given W E C2(JR+), a concave function, the entropy is defined for non negative functions h by
SIJ!(h) = 1 W[h(>.)]d>..
The spectral density h* which maximizes an entropy among the solu tions of (1) or (2) corresponds to a process X* with particular properties. For the Burg entropy, where W(h(>')) = logh(>'), the distribution of X* is the most regular possible, for the Kullback (or Shanon) entropy, where W (h(;\)) = h(;\) log h(;\), it is the closest to a given one ...
Methods of realization of moment problems with entropy maximization 23
This probabilistic point of view leads to results such as developed in Part 1 for the entropy Slog.
In Part 2, we use a Hilbertian setting to construct a rational function solution of (2).
Its denominator is the prediction polynomial. If X admits a spectral representation (3), the prediction polynomial gives the linear predictor of Xo given (XkheK\{O}, optimal in quadratic mean.
The construction is based on the projection of 4>0 onto PK\{O} for thed scalar product and on its connection with the usual scalar product of L~(I). We are interested here in showing that its practical application comes down simply to solve linear systems. Note that it also yields a fixed point method to construct a rational function solution of (1), with a chosen numerator. When it is chosen to be constant, the solution maximizes the Burg entropy.
1. Probabilistic point of view
We gave in [4] the form of the density h* maximizing an entropy functional. We shall give here some of its probabilistic consequences. Let us first
recall it.
Theorem 1 Let 1J~ be the set of nonnegative solutions of (1). Assume there exists a positive h * E V~. Then
SIJ/(h*) = max SIJ/(h) if and only if ho = !(qT/)-l 0 g, g E PK. he1J'J, P
In certain cases, it is possible to get rid of the existence assumption. Theorem 2 Let r = 1 or 2. Let (> be a family of continuous functions. Then a polynomial 9 = EkEK gk4>k in PK exists such that h* = 1/g E V~. Slog is maximum over 1J~ at h*.
Proof Dacunha-Castelle & Gamboa [3] proves that agE PK exists such that 1/g is solution of (1) and is locally integrable. But if r =1 or 2, this is true only if 9 never takes the value 0 over I. And Theorem 1 yields the result. 0
This theorem has a probabilistic consequence which we give here for r = 1.
Corollary 1 Among all stationary processes with given autocovariance co efficients (co, ... ,cn), there exits an AR process. This process gives the max-
imum of Slog and its spectral density is h*(>.) = En 1 k)". k=O ak cos
This can be extended to processes having a representation of the form (3). It can also be generalized to problems combining (1) and (2) such as the determination of processes with given autocovariances and impulse response
24 VALERIE GIRARDIN
coefficients. For r = 1, this problem is equivalent to the determination of h E L1 (11) analytic such that
{ 2~ lr eikA lh(A)1 2dA = ci, 0 ~ j ~ n,
2~ Z eikAh(A)dA = dk, 1 ~ k ~ m.
Franke [2] proved that Slog is maximum among the solutions at the den sity of the ARMA process, that is for a trigonometric rational function. Extensions of this result for multidimensional processes admitting a spec tral representation (3) will be given in a future paper, Castro & Girardin [1]. They lead to the construction of generalized rational functions such as made in Part 2.
Optimization results too can be developed, such as the following which is a direct consequence of the preceding propositions.
Theorem 3 Let tP E L1(I). Let c E [-1,1]. Let 1'(1,4» be the set of non negative functions h E LHI) such that II h(A)dA = 1 and II htP(A)dA = c. Then,
sup sup Slog(h) = 1Iogh*(A)dA, forh*(A) = l+a1cosA 4>eL~ (I). he'D(l .• ) 1
This means that the maximum of the Burg entropy for densities under the constraint II h4>(A)dA = c is obtained for the density of the AR process.
2. Construction of a rational function
We construct here first a rational function solution of (2). Its form is given in the following theorem, which characterizes also the prediction polynomial
Pmin' We ca.ll rational functions the quotients of elements ofPKePK. We have
the orthogonal decomposition
Let (Pk)keL be an orthonormal system for the usual scalar product of L~(I) suited to this decomposition such that (Pk)keK is deduced from (tPk)keK starting by Po constant as 4>0. Let AK be the change matrix from (tPk)kEK to (PklkeK' Theorem 4 Let D = (dkl)(k,l)eKxK be a positive definite matrix. Then,
the projection - P min for the d-scalar product of Po onto PK\{O} is
Methods of realization of moment problems with entropy maximization 25
Moreover, i!(Po+Pmin)(>'):f 0 over I, there exists Q E (PKePK)8PK such that h = (lIPo + Q)/(Po + Pmin) satisfies (2). The proof, given fully in [4], is based on the following idea: lIPo is the pro jection for the d-scalar product of Po + Pmin onto PK. Set! E PKePK such that < <Pk,<Pt >d=< !<Pk,<P/ >q(I) for (k,l) E K x K. Then lIPo is the projection for the usual scalar product of !(Po + Pmin) onto PK. Thus 1I and Pmin are determined. Then h is taken such that its projection for the usual scalar product onto (PKePK)8PK is equal to the same projection of ! - lIPo/(Po + Pmin).
Let us show its practical form in the algebraic case. The product of two elements of PK belongs to PK+K, where K+K={l + 1'/1 E K, l' E K}. Therefore, a measure realizes the d-scalar product (2) if its density h is so lution ofthe moment problem (1) for <Pk(>') = >.k, k E K+K and Ck = d/+11. Note that here D is a definite positive matrix if (Ck)kEK is a positive defi nite sequence and that h is here a rational function in the usual sense.
Take for example r = 1, p equal to A over I = [-1,1] and K = {0,1}. Then! = c(jPo + ci PI + ciP2, where c* = Akc and
bO = 0 ~ 0 . 00) (~ 00)
o e2 ;jl 0 ~
Here, Pmin = b1Ph and the projection of !(Po + Pmin) onto PK is
( < ! Po, Po > <! Po, PI > ) ( 1) (1I) < ! Po, PI > <! Ph PI > b1 0'
so that b1 = -acI/bc2 and 1I = a2(co - CUC2). Then h = (lIPo + Q)/(Po + b1Pt}, where Q = b2P2 is given by
11 h(Po + Pm in)P2(>')d>' = c2. -1
Hence b2 = [aK2(coK - lIH)]/[(eoK2 + e2)H - 2e2K], where K = -Cl/C2
and H = In[(1 + K)/(1 - K)], and finally
h(>') = (all + b2eo + b2e2>.2). (a + aK >.)
If 0 < leI I < c2, h is well defined and its denominator is nonnegative over I. Its numerator too is nonnegative over I if f32/1l belongs to the in terval [-a(eo + e2)-1,-aeo1] or [1s,ts]. If Co = 1 (corresponding to a
26 VALERIE GIRARDIN
probability measure), this is generally true for C2 - leI I et 1- C2 both small enough or both large enough.
This construction yields a fixed point method for the determination of
a rational function h = I/fo°: j solution of the moment problem (1)
whose numerator is chosen and whose denominator is a polynomial of PK. Indeed, Q is above implied by the process of construction from 1/ and Pmin '
If the moments are only given for k E J(, this can be reversed, Q implying 1/ and P. Hence, a linear system may be written whose unknowns are the coefficients of P. Its matrix involves the known moments c of the solution h and its now unknown moments a for ti>kti>/. The determination of the fixed point of the function connecting a to the moments of h constitutes an iterative process for the determination of h.
If Q is chosen equal to 0 over I and p equal to A, then the rational function obtained by this fixed point method is of the form
1 h= ,
EkEK akti>k
and maximizes the entropy Slog among the solutions of (1). Note that in the trigonometric case (i.e. for stationary processes), the
denominator of h can be expressed as the square of a polynomial, a more usual form. A Hilbertian setting was used in Landau [6] too to construct solutions of the trigonometric moment problem for r = 1.
References
1. G. Castro & V. Girardin Multidimen!ional nondationary proceues and entropy maximization In Preparation
2. J. Franke ARMA proceue! have maximal entropy among time series with prescribed autocovariances and impulse re!pon!e! Adv. Appl. Prob., V17, p810, 1985
3. D. Dacunha-Castelle & F. Gamboa Maximum d'entropie et probleme de! moments Ann. Inst .H. Poincare, Prob. Stat., V26, p567, 1990
4. V. Girardin Methodes de realisation de produit !calaire et de probleme de moments avec maximi!ation d'entropie Stud. Mat., 1996
5. M. Krein et A. Nudelman The Markov moment problem and extremal problems Amer. Math. Soc., 1977
6. H. Landau Maximum entropy and the moment problem Bulletin of the American Mathematical Society, V16, 1987
7. M. Priestley Non linear and nonstationary time series analysis Academic Press, London, 1988
MATRICES OF HIGHER MOMENTS:
SOME PROBLEMS OF REPRESENTATION
E. KAARIK Institute of Mathematical Statistics University of Tartu 2 J.Liivi, EE2400 Tartu, ESTONIA
1. Introd uction
There exist different representations of matrices - via tensors, sets, block matrices, vectors. Each of these representations has some preferencies and some shortages in different procedures. In multivariate statistics some addi tional problems arise due some kind of symmetry of the arrays, causing the identity of some groups of elements. It is reasonable to find a parsimonious representation of multivariate matrices, avoiding repeated elements caused by symmetry of the array. In the paper the compressed representation for symmetrical multi-dimen sional arrays (Tiit, Tammet, 1994) will be introduced. The results will be applied for the special case, when a matrix of empirical moments as the array is regarded, and some estimation algorithms will be given (Kaarik, 1994; Kaarik, Tiit 1995), using the classification of moments by their types.
2. Index vectors, index-sets and arrays
Let us start with the definitions of the basic concepts. Definition 1. The vector I = (il' ... , ik) is said to be an index-vector, if
all its components are integers, ij ~ Pj, j = 1, ... , k, where (Pl,'" ,Pk) is a fixed vector. The natural number k is the cardinality of I. If the condition ij < ij+l or ij ~ ij+l is satisfied for j = 1, ... , k - 1, then the index-vector I is said to be ordered (partially ordered). In the case of opposite inequalities the index-vector is D-ordered or partially D-ordered (decreasingly). If all parameters Pj are equal, Pj = p, j = 1, ... , k, then the index-vector I is
27
V. Benes and J. 5tepan (eds.), Distributions with given Marginals and Moment Problems, 27-33. © 1997 Kluwer Academic Publishers.
28 E. KAARIK
said to be symmetrical and p is said to be the order of index-vector. The set of all symmetrical index-vectors of order p and cardinality k will be denoted by I(p, k).
Definition 2. Let us have a set of elements ail ... ik' depending on an index-vector I = (il, ... ,ik), where 1 ~ ij ~ Pj, j = 1, ... ,k. Then we say that we have a k-dimensional array A. In some cases we assume that the symmetricity conditions are fulfilled, as well:
Definition 3. The array A with elements ail ... ik is said to be symmetric, if the following conditions are fulfilled: 1) All indices have the same set of values, Pj = p, j = 1, ... , k,
i.e, Ie I(p,k). 2) The elements ail ... i k are exchangeable by the indices. Almost all useable in practice statistical functions (empirical or theoreti cal) of k random arguments, calculated by p-variate random vector (e.g. moments, central moments, cumulants and other statistics) satisfy the con ditions 1 - 2.
3. Compressing transformations
3.1. EQUIVALENCE CLASSES OF INDEX-VECTORS
Let us regard the symmetric index-vectors IE I(p, k), where the parameter p and the dimension k are fixed. Let us define the equivalence classes of index-vectors by permutations of indices in the following way:
Definition 4. Let I = (ill" .,ik) and J = (jl, ... ,jk) be two index vectors from the class I(p, k). If there exists a permutation P so, that the equality I = P J holds, then we say that the index-vectors I and J are equivalent. Shortly, the index-vectors, corresponding to the same index-set, are equiv alent. As a representative of a class of equivalent index-vectors we will take the partially D-ordered index-vector l' = (i~, ... ,i~J from this class, where
., ., . 2 k tj ~ tj_l, J = , ... , . (1)
The index-vectors that belong to different classes are said to be essentially different. In the future we will regard the set ili(p, k) of equivalence classes (essentially different index-vectors).
3.2. ORDERING OF INDEX-VECTORS
Let us define the position f of a component if of an index-vector I as its successive number in the index-vector (ib"" ik)' Every symmetric index vector (i1 , ... , ik) can be regarded as an k-figure natural number in the
MATRICES OF HIGHER MOMENTS 29
positional number system having the basis p. Notice that for getting the usual form of natural number we should take the transformation ij = ij - 1, j = 1, ... ,p. Let us order all index-vectors ofthe set 'II(p, k) lexicographically; as a result we receive the sequence £(p, k) with terms h (here h is the h-th element ofthe sequence). The sequence £(p,k) = {h} does not include all possible k-figure p-numbers, but only these which satisfy the conditions (1).
Example 1. Let us regard the set of fourth moments of a five-variate random vector i = (Xl,X2,X3,X4,XS)' The usual way to present the set of moments is to use the moment's matrix (of order 5 x 5 x 5 x 5). In this matrix, which is usually written with the help of 5 x 5 blocks, there are 625 entries. But most of the elements of the matrix are superfluous, as almost all moments are represented in this matrix repeatedly. The situation is similar to that of correlation matrix, where all correlation coefficients are represented twice: in the upper and in the lower triangle. The main practical problem is to eliminate the repeated copies of the mo ments from the matrix. The order of moments k is dimensionality of the index-vector and the dimension of random vector p is the order of index vector. It is possible to get all different fourth moments of the random vector, identified by the indices, as a finite sequence £(p, k). In Table 1 all different fourth moments of a five-variate random vector are represented as 4-dimensional index-vectors, identifying the indices of random variables (the sequence £(5,4)).
TABLE 1.
h(I) h(I) h(I) I h(I) h(I)
1 1111 15 3333 29 4431 43 5331 57 5521 2 2111 16 4111 30 4432 44 5332 58 5522 3 2211 17 4211 31 4433 45 5333 59 5531 4 2221 18 4221 32 4441 46 5411 60 5532 5 2222 19 4222 33 4442 47 5421 61 5533 6 3111 20 4311 34 4443 48 5422 62 5541 7 3211 21 4321 35 4444 49 5431 63 5542 8 3221 22 4322 36 5111 50 5432 64 5543 9 3222 23 4331 37 5211 51 5433 65 5544 10 3311 24 4332 38 5221 52 5441 66 5551 11 3321 25 4333 39 5222 53 5442 67 5552 12 3322 26 4411 40 5311 54 5443 68 5553 13 3331 27 4421 41 5321 55 5444 69 5554 14 3332 28 4422 42 5322 56 5511 70 5555
30 E. KAARIK
Let us assign to every index-vector h its rank h = h(I). Our task is to find the ordering rule </>, describing the correspondence h ~ h(I) and </>-1: h(I) ~ h. The ordering rule will be given by the following theorem (Tiit, Tammet, 1994)
THEOREM 1. Let W(k,p) be the set of essentially different index vectors (i1, ... ik), where the dimensionality k and the parameter pare fixed. The rule </> for the lexicographical ordering of index-vectors in the sequence £(p, k) is the following:
k
h(I) = 1 + L A~g+-Lg, (2) g=l
h I ( . .) d AP (p+k-1)(p+k-2)· ... ·p were = t}, ... , tk an k = k! .
The rule </> is a generalization of the well-known operator vech, transforming the upper triangle of a symmetric matrix into the vector. From Theorem 1 immediately two corollaries follow.
COROLLARY 1. The cardinality of the sequence £(p,k) is A~. COROLLARY 2. The transformation </> has a unique inverse transfor
mation </>-1. The rule </> and its inverse </>-1 are both easily programmable.
4. Classification of higher moments of a random vector
In the set W(k,p) there are different types of fourth moments, depending on the number of variables, included into the moment and the degrees of them. To describe the higher moments of a random vector it is convenient to use the concept of the partition of a number, see e.g. Andrews (1976). Let us regard the set of moments of order k of a p-dimensional random vector, k ~ p, and introduce the following classification, using the concept of the partition of a number.
Definition 5. Let n be a natural number. A set of natural numbers Q = {Q1, ... , qv}, satisfying the condition
is a partition of the natural number n, the addendae qj are said to be the parts of n, v = K,( Q) is the cardinality of the partition Q.
Usually the set Q is represented as a non-decreasingly ordered set of parts. As the parts have the values 1,2, ... , n, so there exists an equivalent representation ofthe partition via its characteristic numbers aj, i = 1, ... , n
in the following sense.
MATRICES OF HIGHER MOMENTS 31
Definition 6. Let Q = {ql, ... , qv} be a partition of a natural number n. Let aj denote the number of repetitions of the integer j in the set Q, i.d., the following equation must be valid:
n
n=I:>.aj. j=l
The numbers aj are said to be the characteristic numbers of the partition Q, and the vector a = (aI, ... an) is its characteristic vector.
Definition 7. Let (Xl,"" Xp) be a given random vector and Q = (ql, .. . , qv) a partition of a natural number k, k ~ p. Then we say that the moment E(X?ll ..... X iq:) belongs to the type Q, if ib"" iv are differ ent integers from the set {I, ... , p}. The following problem is to clear up the number K,k(Q) of moments of a type Q in the sequence £(p, k) . The answer to this problem will be given in the following theorem.
THEOREM 2. Let £(p, k) be a sequence of all moments of the order k of a p-variate random vector, and let Q be a partition of the number k, having the characteristic vector a = (al, ... ,ak), see Andrew (1976). Then the number of moments of the type Q equals to the following expression:
k
K,k(Q) = II C;;, (3) j=l
where: dJ = p - bj - 1 , bj = 2::::~=1 aj, bo = 0; j = 1, ... , k and C: = n!j(m!· (n - m)!).
Example 2. Let us regard the set of the fourth moments of a five-variate index-vector, given in Example 1. As the number 4 (the order of moments k) has 5 different partitions ([4], [3,1], [2,2], [2,1,1], [1,1,1,1]), so there are five different types of the fourth moments, see Table 1. To the partition [4] all marginal moments correspond, to the partition [2,2] the moments E(X? . Xi), etc. Let us calculate the cardinalities of different fourth moments of all types K,4(Q). At first, let us write the characteristic vectors of all partitions:
[4]::} (0,0,0,1); [3,1]::} (1,0,1,0); [2,2]::} (0,2,0,0); [2,1,1]::} (2,1,0,0); [1,1,1,1]::} (4,0,0,0);
The cardinalities are calculated using the formula (3): K,4{[4]} = CJ = 5; K,4{[3][1]} = CJ . Cl = 20; 1'1,4 {[2][2]} = Cg = 10; 1'1,4 {[2][1][1]} = Cg . C~ = 30; K,4{[1][1][1][1]} = ct = 5;
It is easy to check from the Table 1 the correctness of results.
32 E. KAARIK
For some types of moments it is quite easy to find the rule for determining their ranks in the sequence £(p, k). E.g., the marginal moment, correspond ing to index j, has the rank h(f) = At, j = 1, ... ,p.
4.1. THE RATIO OF NUMBER OF DIFFERENT MOMENTS AND NUMBER OF ALL MOMENTS
THEOREM 3. The ratio of number of different moments and number of all moments of the order k of p-dimensional random vector approaches to the constant b as p tends to infinity, i.e. the following equality is valid
1. A~ 1 1m -k = k' . p-oo p .
(4)
Example 3. Let us see the number of all4-order moments (p4), the num
ber of different moments (A~) and the ratio (#) by different dimensions p p of random vector
TABLE 2.
AP 4 5 15 35 70 126 210 330
p4 16 81 256 625 1296 2401 4096 ratio 0.312 0.185 0.136 0.112 0.097 0.087 0.080
By connection (4) the limit of the ratio is ~ = 0.0417. Example 4. Construction of a matrix of fourth moments of a random
vector having normal distribution and constant correlations. Let us see the set of fourth moments of a bivariate random vector X = (XI, X 2 ), X rv N2(0, R), DXi = 1. Different fourth moments of the type Q, representative index-vectors and all possible index-vectors are: Q = {4}, 1-'4 = 3; (1,1,1,1),(2,2,2,2); Q = {3,1}, 1-'31 = 3p; (2,1,1,1,) -+ (1,2,1,1),(1,1,2,1),(1,1,1,2);
(2,2,2,1) -+ (2,2,1,2),(2,1,2,2),(1,2,2,2);
Q = {2,2}, J.l22 = 1 + 2p2; (2,2,1,1) --+ (2,1,2,1),(2,1,1,2),(1,2,2,1), (1,2,1,2),(1,1,2,2).
Compressing representation (representative index-vectors) of fourth mo ments of a two dimensional random vector is: 1. (1,1,1,1) 3 2. (2,1,1,1) 3p 3. (2,2,1,1) 1 + 2p2 4. (2,2,2,1) 3p 5. (2,2,2,2) 3 As we can see, the compressing representation is a good tool for simpli fying the representation procedures and calculation of matrices of higher moments.
References
Andrews, T.W. (1976) The Theory of Partitions. Kiiiirik, E. (1994) Representations of index-functions of random vector. Ma&ter-thesis. Kiiii.rik, E., Tiit E.-M. (1995) Arrays of Multivariate statistics and their representation,
New Trends in Probability and Statistics, Vol. 3, Multivariate Statistics and Matrices in Statistics, pp 313-324, Vilnius, Utrecht, Tokyo.
Tiit, E.-M., Tammet, S.{1994} Using Index-Vectors and Partitions in Multivariate Anal ysis, Acta et Comment UT, Vol. 968, pp 55-75
THE METHOD OF MOMENTS IN TOMOGRAPHY AND
QU ANTUM MECHANICS
AND
RACHEV S.T. University of California Santa Barbara Santa Barbara, CA 93106-3110, USA§
1. Radon transform and its applications to computer tomogra phy
Suppose an x-ray examination of an object (a body) is performed, and the intensities of the x-ray at both input (1d and output (10) are measured. It is known, that (under some conditions imposed on the object and the intensity of the input x-ray) the following is true:
L j 10g(...2.) = p(x)dL, fo L
where the integral is taken along L, the straight line which the x-ray follows through the body. The problem is to reconstruct the density p of the body based on the line integrals calculated on (some or all) of the straight lines.
The transformation from the density p( x) to the set of all its line in tegrals (considered these as functions of the parameters of the line) is the so-called Radon transform. This was introduced by Radon in 1917 [13], who also provided formulae for the inversion of the transform and proved the uniqueness of the reconstruction of the density from its Radon transform.
IThe research was supported by RFFI Grants No 96-01-00852 and 95-01-01260 §The research was conducted while S.T. Rachev was visiting the University of Freiburg
and Kiel. His visits were supported by a Research Award for U.S. Senior Scientists from the Alexander-von-Humboldt Foundation.
35
V. BeMI and J. Stlptin (eds.), Distributions with given Margi1llJis and Moment Problems, 35-52. @ 1997 Kluwer Academic Publishers.
36 KLEBANOV L.B. AND RACHEV S.T.
In fact, by the Radon transform any probability measure in the n - dimensional Euclidean space can be reconstructed in a unique way in terms of probabilities on half spaces. Probabilists refer to this result as the Wold Cramer principle; has been obtained independently by Wold and Cramer in 1932, using characteristic functions.
In the medical sciences applications of Radon's result began in the six ties with the first tomography machine of A. Cormack and the first use of the commercial tomography machine by G . Hounsfield; they were both awarded the Nobel prize in Medicine in 1979 for this work.
Computer tomography (CT) is based on the inversion of the Radon transform, which allows us to reconstruct in a unique way the density of a measure.1 However such a unique reconstruction is possible only if one knows an infinite number of marginals of the underlying distribution. In practice one can only have a finite number of marginals, and a unique reconstruction of the density is impossible in this case. One can easily see this from the following "CT paradox".
Let I( x) be a probability density function given on Euclidean plane R2 and having a compact support D C R2. Let 01 , ••• , ON be N directions (N unit vectors) in the plane R2. It is obvious, that a measure on R2 of compact support is uniquelly determined by infinitely many marginals.
It was proved in the recent work [3], that for any density I: 0 ~ I(x) ~ 1, xED and for any finite number of directions 0], ... , ON (N < 00) there exists another density lo{ x), xED such that 10 has the same marginals in the directions 01 , • •• ,ON as 1 and such that 10 has only two values 0 or 1. This result gives the following CT Paradox (see (3)):
It implies that for any human object and corresponding projection data there exist many different reconstructions, in particular, a reconstruc tion, consisting only of bone and air (density 1 or 0), but still having the same projection data as the original object. Related non uniqueness results are familiar in tomography and are usually ignored because CT machines seem to produce useful images. It is likely that the "explana tion" of this apparent paradox is that point reconstruction in tomogra phy is impossible.
In the first part of this paper we give a survey of some recent already published results, conencted with CT Paradox. The second part is con cerned with an analogous in quantum mechanics for wich new results are obtained.
IThe methods of computer tomography CT (see for example the survey [14], and books [11], [12]) have been used in science, medicine and technology.
THE METHOD OF MOMENTS IN TOMOGRAPHY... 37
2. The problem of "stability" in computer tomography with par tially known data
The CT paradox shows that it is impossible (in general situation) to re construct the density basing on partially known tomography data. The purpose here is to show that, under moment-type conditions, measures having a "large" number of coinciding marginals are close to each other in metrics metrizing the weak topology (so-called here "weak metrics"). Our method is based on techniques used in classical moment problem. The key idea in showing that measures with large numbers of common marginals are close to each other in the weak metrics is best understood by compar ing the following three results. The first is the Theorem of Guttman et al. (1991) [3] mentioned above. The second (see [6, page 265]) states that if a finite number of moments /-LI. .•. ,/-Ln of a function f, 0 ::; f ::; 1 are given, then there exists a function 9 which takes the values 0 or 1 only, and pos sesses the moments /-Ll, .•. ,/-Ln. It is clear that these two results are similar; however, the condition of equality of the marginals is more complex than the condition of coincidences of the moments. Finally the third result (see [5, p.170-197]) gives estimates of the closeness in the A - metric on RI (a weak metric) for measures having common moments ILl, ••. , /-Ln (n < (0). These estimates are expressed in terms of the truncated Carleman's series 13m = L~=1 IL~]I(2j), (2m < n), and show that the closeness is of order
f3~1/4. Of course, since the condition of common marginals seems to be more restrictive than the condition of equal moments, one should be able to construct a similar estimate expressed in terms of the common marginals only.
First we shall derive estimates for closeness of measures in R2 having coinciding marginals in n directions. We consider the case when one of the measures has a compact support: in this case, the A-closeness of measures has order lIn. Further, the compactness assumption will be relaxed by Carleman's assumption in the problem of moments. Here the A-closeness of measures is of order f3:N4. We also display estimates of the closeness of measures with f-coinciding marginals, (0 < f < 1). These estimates differ from the corresponding ones with equal marginals by an additional term of order 1Jlog(1/f). We conclude by applying our results to the problems of computer tomography. In particular we offer a solution of the computer tomography paradox mentioned above and compare this solution with anal ogous results in [7J. To highlight the basic ideas, let us consider only the 2-dimensional case in full.
Let (h, ... , On be n unit vectors on the plane and PI, P2 be two prob abilities on R2, having the same marginals in the directions 01 , ••• ,On. To estimate the distance between PI and P2, different weak metrics can be
used; however it seems, that the .x-metric is the most convenient for our purposes. This metric is defined as
(see, e.g., [17]), where (.,.) is the inner product and II." is the Euclidean norm. Clearly, .x metricize the weak convergence.
Our first result concerns the important case where one of the probability measures considered has compact support (see [8]). Theorem 2.1 Let (h, ... , On be n ~ 2 unit vectors in R2, no two of which are collinear. Let the support of the probability PI be a subset of the unit disc, and let the probability P2 have the same marginals as PI in the direc tions OJ, ... , On. Set
S=2[n;1], (2.2)
where [r] denotes the integer part of the number r, then
(2.3)
Remark 2.1 Let 01 , ••• , On be n ~ 2 directions in R2 no two of which are collinear. Suppose that the marginals of the probabilities PI and P2 along the directions 01 , •.• ,On have the same moments up to the even order k :::; n - 1. Then the marginals of PI and P2 with respect to any direction have the same moments up to order k.
Remark 2.2 In view of Remark 2.1 the proof of Theorem 2.1 even implies that Theorem 2.1 still holds if we replace the assumption that PI and P2 have coinciding marginals along the directions OJ (j = 1, ... , n) with the assumption that these marginals have the same moments up to order n - 1.
We next relax the condition of compactness for the support of PI, assum ing only the existence of all moments together with Carleman's conditions for the moment problem.
Set J.Lk=SUP f (x,O)kPI (dx),k=O,l, ... ,
9E81 iR2 (6-2)/2 1
f36 = L J.L;j2), j=1
where the number s is determined from equation 2.2 and SI is the unit circle.
THE METHOD OF MOMENTS IN TOMOGRAPHY.. . 39
Theorem 2.2 [8} Let (h, ... ,8n be n 2: 2 directions in R2 no two of which are collinear. Suppose that the measure PI has moments of any order. Sup pose also that the marginals of the measures PI and P2 in the directions 81 , .•. , 8n have the same moments up to order n - 1. Then there exists an absolute constant C such that
Let us give another type of results (see [7]). Let FD be the set of all probabilistic densities with compact support in
the square D = {x = (X},X2) :1 Xl I~ 1,1 X2 I~ I}.
Let i.p(1 be the Gaussian density:
and let f * i.p(1( x) (J E F D) be the corresponding Gaussian convolution.
Theorem 2.3 Let n 2: 2 be a natural number. There exist N = 2n di rections 81 , ..• ,8N on the Euclidean R2 plane such that if the densities f, g E FD have the same marginals on the directions 81, ... , 8N then
sup 1 f * i.p(1(x) - g * i.p(1(x) I~ 1/(y1ran+22(3n-4)/2r«n+ 1)/2)), (2.4) xER2
where r( z) is the Gamma-function. The vectors 81 , ... , 8N can be chosen on the following definition:
8j = (vj,-I)/(v] + 1)1/2,j = 1, ... ,n;
8j = (l,vj-n)/(V]_n + 1)1/2,j = n+ 1, ... ,2n,
where Vk = cos(1I'(2k-l)/(2n)),k = 1, ... ,n.
Let us now consider a more realistic situation where the marginals of PI and P2 in the directions 81 , ... ,8n are not the same but are close in the metric A. We use the same notation as that introduced in Theorem 1.
Theorem 2.4 Let 81 , ... ,8n be n 2: 2 directions in R2, no two of which are collinear. Suppose that the supports of the measures PI and P2 lie in the unit disc, where they have f - coinciding marginals with respect to the directions 8j (j = 1, ... , n) i.e.,
(2.5)
= min max ( max I 1t'1( Tj 8j) - 1t'2( Tj 8j) I, l/T) $ E, T>O I"I~T
j = 1, ... , n. Then there exists a constant C depending on the directions 8j (j = 1, ... , n) such that for sufficiently small E > 0, we have
1 A(Pb P2) $ G(l/log( -) + l/s),
E (2.6)
where, as before, s is determined from equation {2}.
Remark 2.3 The statement in Theorem 2.4 still holds if instead of the E-coincidence of the marginals we require the E-coincidence of the mo ments up to order s of these marginals. The latter condition is defined by the inequalities
Let us demonstrate another application of the method of mo ments to problems with incomplete data.
Consider a body emitting x-rays. This body can move along horizontal axis in intervals [bo, (0) and (-00, -bo]. These x-rays go through a body situated in the strip {(x, y) : Ixl $ bo,lyl < oo}. H the body has density p(x,y) then we know all integrals of the form J p(x,a(x - b»dx where Ibl ~ bo and a is an arbitrary number. Let us denote
t/J(a, b) = j p(x,a(x - b»dx.
Actually, t/J( a, b) is well defined for all real a and b. Only the case under condition Ibl ~ bo is of interest. More precisely, one is studying the situation where t/J( a, b) is observed for all real values a and b such that Ibl ~ boo So, for any positive integer n we have
100 bnt/J(a,b)db = [00 bndbjp(x,a(x _ b»dx. bo Jbo
After substituting y = a( x - b) we get
100 j 100 y 1 bnt/J(a,b)db = dx (x - - )np(x, y)-dy bo a(z-bo) a a
~ (n) (_l)n j d 100 k n-k ( ) =L.J k n-k+t X xy px,ydy. k=O a a(z-bo}
We can represent the summands in the right hand side of previous equality in the form
But
Therefore
= J dx 1000 xkyn-kp(x, y)dy + O(an- k)
and
[00 bn1/1(a, b)db = t (:) (:':~l J dx [00 xkyn-kp(x, y)dy + O(a-1). Ao k=O a h
Hence, knowing the function 1/1 implies that we also know all integrals
J dx 100 xkyn-kp(x, y)dy
for all n = 1, ... and k = 0, ... , n - 1. In other words, we know all moments of the density (as the coefficients of expansion in powers of ~)
Pl(X, y) = y2p(x, y).
In case when the density p(x, y) has compact support (or in case when the measure is defined by its moments in unique way) we can reconstruct the density Pl(X, y) in unique way by its moments. It means that we can reconstruct the density p( x, y) in unique way for y i: O.
3. Quantum mechanics and computer tomography
Consider a physical system with one continuous degree of freedom, with a generalized coordinate operator x and a conjugate operator p. Suppose that "the rotated quadrature" operators xe, pe are defined by
Xe=xcos8+psin8, pe=-xsin8+pcos8 (3.1)
for all angles 8 (xo, Po are related with X, p (9 = 0) by unitary trans formations). Let Po( xo) be the probability distribution of the observable
Xo. It was shown in [16] that if an uncountable set of such distributions {Po(xo) is known, then the corresponding Wigner distribution W(x,p) for pure state
W(x,p) = ~ J 1fJ(x + x')1fJ*(x - x')ei2PX'dx' (3.2)
can be determined uniquely by the Radon transform (see, for example [11]). W(x,p) is in one-to-one relation to the wave function:
1fJ( x + x')1fJ* (x - x') = J W (x, p )ei2pX' dp. (3.3)
An experimental method in [15] for measuring of corresponding distribu tions Po(xo; W) is given by
Po(xo; W) = J W(xo cos 8 - Po sin 8, Xo sin 8 + Po cos 8)dpo; (3.4)
Po(xo; W) represent the set of marginals of W(x,p)2. So to "measure" of the wave functions one needs to solve the inverse problems: (A) Po(xo; W) - W(x,p) and (B) W(x,p) - 1fJ(x) . To solve (A) via Radon transform the filtered back-projection numerical algorithm (FBPNA) of the usual tomography [11] was used in the orig inal works [15]. Both inverse problems (A) and (B) are ill-posed (in the Hadamard sense, see for example [2]) mathematical problems, what means that very small errors in Po( Xo; W) may cause very large (or even infinite) errors in W(x,p), and correspondingly, in wave functions. Moreover, in fact we can only measure a finite number (N) of marginals Po(xo; W), 8 E {81 , ... ,8N}; however for a finite number of marginals the Radon Theo rem on the uniqueness of solution of the inverse problem (A) fails. As it was recently derived [3], this non-uniqueness produces the so-called tomog raphy paradox. In our terminology this means that two different Wigner distributions W1(x,p) and W2(x,p), which have the same N marginals, may differ essentially. Another disadvantage is that Wigner distributions W(x,p) cannot have bounded support. In fact,
11fJ(x)1 2 = J W(x,p)dp, 1..j;(p)12 = J W(x,p)dx (3.5)
where ..j;(p) is the wave function in the momentum (impulse) representation
-¢;(p) = J 1fJ( x )e -ipx dx. (3.6)
2In general, the Wigner distributions W(x,p) may not be measurable in physical experiment since W(x,p) is not non-negative in general. It was proved in [4] that W(x,p) are non-negative iff the wave functions are Gaussian.
If we assume that W(x,p) has bounded support, then (3.5) implies that tP( x) and ;j,(p) both must have bounded support, but this is impossible due to the Paley - Wiener Theorem. Therefore in reality only a finite number of truncated marginals F9(X9j W) are available, say
(3.7)
w here the finite X 01, X 92 are determined by the possibilities of the exper imental device. It follows from the finiteness of the number of truncated marginals F9( X9j W), (J E {(JI, .. . ,(IN}, and necessary discretization of the integral Radon transform [11] that the solution of problem (A) produced by FBPN A [15] has the so-called artifacts. Artifacts are non-existing details in the exact W(x,p) and tP(x) which originate from the non-accuracy of mathematical part of the original scans. Such artifacts are easy to detect even in the simplest case of the Gaussian W(x,p), see [15].
Using the methods in [7] and [8] we obtain the following result. Its proof goes beyond the scope of this paper and will be published elsewhere.
Theorem 3.1 Let tPI, tP2 be two wave functions and W}, W 2 be the corre sponding Wigner distributions. Suppose that tPl (I = 1,2) has the modulus of continuity w( u) and
ItPI(x)1 S; c}, 1 ItPI(x)ldx s; c2,
gl(a,(jO') = 271'10'21 1 tPI(a-(-6 -6)-l/Ji(a-6)
exp( -(~i + ~i)/(20'2))d6d6, (l = 1,2), (3.8)
where C 1, C 2 are positive constants. For any f > 0 and any integer N ~ 2 there exist N + 1 directions (Jo, ... ,(IN such that if the marginals of WI and W2 in these directions are f-identical, i.e.
(3.9)
suplg1(a,(jO')-g2(a,(jO')1 S; (3.10) a
where C is absolute constant. Moreover, the directions 80 , •.. , 8N may be chosen so that the points
1 1 ao = - tan 80 ' •.• ,aN = - tan 8N
are dividing the interval [-A, A] into N equal parts.
Remark 3.1 The condition (3.9) may be regarded as a measure of the deviation of the truncated from the exact marginals.
Remark 3.2 In (3.8) the Gaussian kernel was used only as an example of a smoothing kernel, but it is possible, indeed, to use other kernels which would lead to similar estimates.
Remark 3.3 For mixed states the estimates for the density matrix can be derived by the same method. The corresponding results will be published separately.
4. A sufficient condition for the solution uniqueness of the moment problem for quasi-probability Wigner distribution
For a given a sequence of constants J.'j,k where j, k are multi-indices r
) = (il'''',)n), k =\(kb ... ,kn), (4.1)
and n > 0 is some fixed integer, Narcowich [9] (see also [10]) provided necessary and sufficient conditions for the existence of a quasi-probability Wigner density W( q, p), (q = (ql, ... , qn), p = (PI, ... , Pn)) such that
(4.2)
(4.3)
The conditions for uniqueness of the problem (4.2) are unknown,3 and the essence of this section is to -give a sufficient condition for uniqueness of the problem (4.2) under assumption of its solvability (that is under conditions derived in [9]). For simplicity we sha.ll consider only the case n = 1, but our results are valid for arbitrary n ~ 1. Note that the problem of moments for W will be equivalent to a system of "probabilistic" problems of moments.
3Narcowich [10] provided an example of non-uniqueness of problem {4.2}.
The Wigner function (more precisely, the quasi-probability Wigner den sity function) of a quantum system in a mixed state is given by the formula:
W(q,p) = ~ f [.l "pj(q + q')"pj(q - q')dF(j)] e2inpql dq'. (4.4)
Complex-valued functions "pi are supposed to be continuous and square module integrable over the whole real line. The symbol * is stated for complex conjugation. The functions "pi are indexed by j from an abstract set J, and F is a measure on J.
Let us formulate some results by F.J.Narcowich. Firstly give the defi nition of Ii-positive definiteness. Let r = Rn x Rn be phase space. Denote z = (q1, ... ,qniP1, ... ,Pn) E r and define
n
= z'Jz,
where
J - (On In) - -In On . Definition 4.1 A continuous function F given on r is called Ii-positive definite if for any integer m 2: 1, for any set {a1, ... , am} C r and any set of complex numbers {All ... , ).m} holds the inequality
m
L F(ai - ak)ei~(7(a",aj)Xi).k 2: o. j,k=l
Let us note that for the case Ii = 0 we have well known definition of ordinary positive definiteness.
Theorem 4.1 (Narcowich) A continuous function F given on r is a Wigner function (4.4) if and only if it is Ii-positive definite.
This Theorem is thus very similar to the well known Bochner Theorem. We suppose here that the Plank constant Ii equals to 1 in the chosen
units system. Now we give some notations (due to Narcowich) connected with the conditions of solvability of the problem of moments for Wigner distribution. Suppose that A and B are two (sufficiently smooth and quickly decreasing at infinity) functions given on the phase space r. Define the operation 0 by
Set also
< A, B >= J A(z)B(z)dz.
Definition 4.2 We shall say that a function F given on r is Ii-positive if
< F,AoA >~ 0
for all A E S(Rn), where S(Rn) is Schwartz space on Rn. For any given polynomial P(z) = L,j,k Cj,kpiqk we define a "moment func tional" J.t:
J.t( P) = ~ Cj,kJ.tj,k j,k
Pj,k = J J piqkW(q, p)dnq dnp,
and then
J.t(P) = J P(z)W(z)dz =< P, W > .
Theorem 4.2 (Narcowich) For the existence of a Wigner function W hav ing prescribed moments J.tj,k it is necessary and sufficient that J.t( P) ~ 0 for any Ii-positive polynomial P. Of course, this Theorem is similar to the well known necessary and sufficient condition for the solvability of classical moment problem.
Suppose the Wigner function has all moments
J.tm,n = J J qmpnW(q,p)dqdp, m ~ 0, n ~ O. (4.5)
Conversely, let {J.tm,n, m ~ 0, n ~ O} be a double sequence of numbers sat isfying the Narcowich conditions. Then there exists at least one Wigner function W for which (4.5) holds. We are interested under what additional conditions posed on the sequence J.tm,n this Wigner function is unique. A partial answer is given by the following theorem.
Theorem 4.3 Given {J.tm,n, m ~ 0, n ~ O} let
and suppose that the series
ji,k = m~ lJ.tk-j,jl J
(4.6)
(4.7)
diverges. Then there exists no more than one Junction W oj the Jorm (4.4) with continuous and square module integrable functions"pj satisfying (~.5).
Proof. Suppose that there exists a function W satisfying (4.5). For arbi trary real a and integer k ~ 0 let us consider the quantity
j j W(q,aq + b)bkdqdb.
Denote
On the other hand in view of (4.4),
J j W(q, aq + b)bkdqdb
( 4.8)
(4.9)
= ~ j j j bk [.l 'lj;j(q + ql)'Ij;j(q - ql)dF(j)] e2i(aq+b)q'dqdbdq'
= 2~ J J J bk [.l 'lj;j(U)'Ij;j(V)dFU)] ei(a;2+bu)e-W~2+bv)dudvdb
where
(f;j( b; a) = j 'lj;j( U )ei('1~2 +bu) duo
From (4.8) and (4.10) we get
Pk(a) = 2~ J bk [J; l{f;j(b;a)12dF(j)] db.
( 4.10)
(4.11 )
Therefore for arbitrary fixed a the sequence {Jlk(a), k ~ O} is a sequence of moments corresponding to a non-negative density
(4.12)
The Carleman's criterion (see, for example,

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Distributions with given Marginals and Moment Problems