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Dissertationes Mathematicae (Rozprawy Matematyczne) POLSKA AKADEMIA NAUK, INSTYTUT MATEMATYCZNY DISSERTATIONES MATHEMATICAE (ROZPRAWY MATEMATYCZNE) KOMITET REDAKCYJNY ANDRZEJ BIALYNICKI-BIRULA, BOGDAN BOJARSKI, ZBIGNIEW CIESIELSKI, JERZYLO S,´ ZBIGNIEW SEMADENI, JERZY ZABCZYK redaktor, WIESLAW ZELAZKO˙ zast¸epca redaktora CCCLVIII JOLANTA K. MISIEWICZ Substable and pseudo-isotropic processes Connections with the geometry of subspaces of Lα-spaces W A R S Z A W A 1996 Jolanta K. Misiewicz Institute of Mathematics Technical University of Wroc law Wybrze˙ze Wyspia´nskiego 27 50-370 Wroc law, Poland E-mail: [email protected] Published by the Institute of Mathematics, Polish Academy of Sciences Typeset in TEX at the Institute Printed and bound by & PRINTED IN POLAND c Copyright by Instytut Matematyczny PAN, Warszawa 1996 ISSN 0012-3862 CONTENTS I. Introduction ........................................................................ 5 II. Pseudo-isotropic random vectors .................................................... 9 II.1. Symmetric stable vectors ...................................................... 9 II.2. Pseudo-isotropic random vectors ............................................... 15 II.3. Elliptically contoured vectors .................................................. 23 II.4. α-symmetric random vectors ................................................... 27 II.5. Substable random vectors...................................................... 32 III. Exchangeability and pseudo-isotropy ................................................ 35 III.1. Pseudo-isotropic exchangeable sequences ...................................... 35 III.2. Schoenberg-type theorems .................................................... 40 III.3. Some generalizations.......................................................... 43 IV. Stable and substable stochastic processes............................................ 45 IV.1. Gaussian processes and Reproducing Kernel Hilbert Spaces.................... 45 IV.2. Elliptically contoured processes ............................................... 47 IV.3. Symmetric stable stochastic processes ......................................... 50 IV.4. Spectral representation of symmetric stable processes.......................... 56 IV.5. Substable and pseudo-isotropic stochastic processes............................ 59 IV.6. Lα-dependent stochastic integrals ............................................. 62 IV.7. Random limit theorems ....................................................... 63 V. Infinite divisibility of substable stochastic processes ................................. 64 V.1. Infinitely divisible distributions. L´evymeasures ................................ 66 V.2. Approximative logarithm ...................................................... 68 V.3. Infinite divisibility of substable random vectors................................. 73 V.4. Infinite divisibility of substable processes....................................... 77 References .............................................................................. 80 Index ................................................................................... 90 1991 Mathematics Subject Classification: Primary 46A15, 60B11; Secondary 60G07, 46B20, 60E07, 60K99. Received 2.3.1995; revised version 18.9.1995. I. Introduction This paper is devoted to a problem which can be expressed in a very simple way in elementary probability theory. Consider a symmetric random vector X = (X1,X2) taking values in R2. For every line ℓ in R2 passing through the origin we define a random variable Πℓ(X) which is the orthogonal projection of X onto ℓ. In other words, Πℓ(X)= e ,X , where e is a unit vector contained in the line ℓ, and , denotes the usual inner h ℓ i ℓ h· ·i product in R2. The problem is to characterize all random vectors X with the property that all orthogonal projections Πℓ(X) are equal in distribution up to a scale parameter. Equivalently, we can say that with every line ℓ is associated a positive constant c(ℓ) such that Π (X) has the same distribution as c(ℓ) X . Random vectors having this property ℓ · 1 are called pseudo-isotropic. The existence of such random vectors is evident: it is enough to notice that multidi- mensional symmetric Gaussian random vectors and multidimensional symmetric stable vectors are pseudo-isotropic. Another example is given by a random vector uniformly distributed on the unit sphere in R2. In this case we can see that the distribution of the orthogonal projection does not depend on the direction of the diameter on which we are projecting. In analytical description of the problem we want to characterize all symmetric random vectors X = (X1,X2) such that their characteristic function ϕX (ξ1t, ξ2t) is the same as the characteristic function ϕ (c(ξ , ξ )t) of the random variable c(ξ , ξ ) X . The X1 1 2 1 2 · 1 function c : R2 [0, ) has to be one-homogeneous. The equivalence easily follows from → ∞ the equality ξ X + ξ X = ξ2 + ξ2 Π (X), 1 1 2 2 1 2 · ℓ where ℓ is a line passing through the origin,q containing the unit vector ξ1 ξ2 2 2 eℓ = , , and c(ξ1, ξ2)= ξ1 + ξ2 c(ℓ). ξ2 + ξ2 ξ2 + ξ2 · 1 2 1 2 q The considered propertyp of randomp vectors can be easily generalized to random vectors taking values in spaces bigger than R2. In the paper we consider pseudo-isotropic random vectors taking values in Rn, pseudo-isotropic sequences of random variables and pseudo- isotropic stochastic processes. Pseudo-isotropic random vectors taking values in infinite dimensional linear spaces are not mentioned in the paper, but the interested reader can find a lot of useful information in the references. By studying pseudo-isotropic random vectors and stochastic processes we do not only want to enrich the class of distributions with properties which are interesting and co- 6 J. K. Misiewicz nvenient for calculations. We also want to propose another method of studying symme- tric stable random vectors and processes which are strictly connected with the idea of pseudo-isotropy. Firstly, the scale parameter c(ξ1, ξ2) can be given by the Lα-norm, for some α (0, 2], i.e. there may exist a linear operator : R2 L (S,Σ,ν) for some ∈ ℜ → α measure space (S,Σ,ν) such that ( ) c(ξ , ξ )= (ξ , ξ ) . ∗ 1 2 kℜ 1 2 kα This means that the characteristic function of a pseudo-isotropic random vector X can be of the form ϕ( (ξ , ξ ) ), while the characteristic function of a symmetric α-stable kℜ 1 2 kα random vector is of the form exp (ξ , ξ ) α . Secondly, representation ( ) holds {−kℜ 1 2 kα} ∗ under a very weak condition on the pseudo-isotropic random vector X: it is enough to assume that there exists ε > 0 such that E X ε < . Thirdly, every known function | 1| ∞ c : R2 [0, ) which can appear in the definition of a pseudo-isotropic random vector → ∞ admits a representation ( ) for some α (0, 2]. ∗ ∈ Historically, people considered 1938, the year that Schoenberg published three papers on spherically symmetric random vectors and completely monotonic functions (see [215]– [217]), as the beginning of the investigation of pseudo-isotropic random vectors. We should remember, however, that even earlier, in 1920’s and 1930’s, Paul L´evy and Aleksandr Yakovlevich Khinchin had introduced the idea of stable random variables and vectors, and also that the beginning of the investigations of Gaussian random vectors goes back to the beginning of the 18th century. Note that spherically symmetric random vectors have a very special property. Namely, both the characteristic function and the multidimensional density function (if the latter one exists) are constant on spheres centered at the origin. This was the reason why the theory developed by Schoenberg broke into two parts. The first one, which we call pseudo- isotropy, describes random vectors and processes having all one-dimensional projections the same up to a scale parameter, which is equivalent with the fixed geometry of level curves for the characteristic function. The second one, only occasionally appearing in this paper, describes random vectors and processes with fixed geometry of level curves for multidimensional density functions and is usually connected with the de Finetti theorem where the exchangeability σ-field is specified by the geometry. The spherically symmetric random vectors and processes were extensively studied and turned out to be very useful in statistics and probability theory. Slightly generalizing the definition to the vectors which are linear images of spherically symmetric random vectors, mathematicians were considering elliptically contoured random vectors and spherically generated random vectors. We shall also mention here that well-known sub-Gaussian random vectors and stochastic processes in both possible definitions (i.e. random vectors which are mixtures of a symmetric Gaussian random vector, and random vectors having all weak moments proportional to the corresponding moments of some Gaussian random vector) are in fact elliptically contoured. More about the history of elliptically contoured random vectors and processes can be found in [44] and [172]. In 1967 Bretagnolle, Dacunha-Castelle and Krivine (see [29]) proved that all positive definite norm dependent functions on infinite-dimensional Lα-spaces are mixtures of cha- racteristic functions of symmetric stable random vectors. This shows also that for α> 2 Substable and pseudo-isotropic processes 7 the only positive definite norm-dependent function on infinite-dimensional Lα-space is constant, so the corresponding pseudo-isotropic random vector is zero everywhere. This paper was important for the theory
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