Infinitely divisible distributions over locally compact non-archimedean fields

This paper is devoted to stochastic processes with values in finite-dimensional vector spaces over infinite, locally compact fields of zero and positive characteristics with nontrivial non-Archimedean norms. Infinitely divisible distributions are studied. Theorems about their characteristic functionals are proved. Particular cases are demonstrated as applications to non-Archimedean analogs of Gaussian and Poisson processes and their generalizations.     

Multi-variable subordination distributions for free additive convolution

Let k be a positive integer and let Dc(k) denote the space of joint distributions for k-tuples of selfadjoint elements in C^∗-probability space. The paper studies the concept of “subordination distribution of μ?ν with respect to ν” for μ,ν∈Dc(k), where ? is the operation of free additive convolution on Dc(k). The main tools used in this study are combinatorial properties of R-transforms for joint distributions and a related operator model, with operators acting on the full Fock space.Multi-variable subordination turns out to have nice relations to a process of evolution towards ?-infinite divisibility on Dc(k) that was recently found by Belinschi and Nica (arXiv: 0711.3787). Most notably, one gets better insight into a connection which this process was known to have with free Brownian motion.     

Infinitely divisible and stable statistical experiments

The purpose of the present contribution to the theory of comparison of statistical experiments is twofold: to describe in a somewhat direct way the functional-analytic approach to the central limit theorem for experiments in terms of infinitesimal triangular arrays, and at the same time to emphasize the application of an indisputable definition of Poisson experiments. Proceedings of the Seminar on Stability Problems for Stochastic Models, Hajdúszoboszló, Hungary, 1997, Part III.     

Random matrix models of stochastic integral type for free infinitely divisible distributions

The Bercovici-Pata bijection maps the set of classical infinitely divisible distributions to the set of free infinitely divisible distributions. The purpose of this work is to study random matrix models for free infinitely divisible distributions under this bijection. First, we find a specific form of the polar decomposition for the Lévy measures of the random matrix models considered in Benaych-Georges [6] who introduced the models through their laws. Second, random matrix models for free infinitely divisible distributions are built consisting of infinitely divisible matrix stochastic integrals whenever their corresponding classical infinitely divisible distributions admit stochastic integral representations. These random matrix models are realizations of random matrices given by stochastic integrals with respect to matrix-valued Lévy processes. Examples of these random matrix models for several classes of free infinitely divisible distributions are given. In particular, it is shown that any free selfdecomposable infinitely divisible distribution has a random matrix model of Ornstein-Uhlenbeck type ?? ₀ ^?? e ^?1 d?? _t ^d , d ?? 1, where ?? _t ^d is a d × d matrix-valued Lévy process satisfying an I _log condition.     

Banach algebras and infinitely divisible distributions

Translated from Matematicheskie Zametki, Vol. 46, No. 4, pp. 60–65, October, 1989.     

The Lebesgue decomposition of the free additive convolution of two probability distributions

We prove that the free additive convolution of two Borel probability measures supported on the real line can have a component that is singular continuous with respect to the Lebesgue measure on ${\mathbb{R}}$ only if one of the two measures is a point mass. The density of the absolutely continuous part with respect to the Lebesgue measure is shown to be analytic wherever positive and finite. The atoms of the free additive convolution of Borel probability measures on the real line have been described by Bercovici and Voiculescu in a previous paper.     

Generalized renewal sequences and infinitely divisible lattice distributions

We introduce an increasing set of classes Γ_a (0?α?1) of infinitely divisible (i.d.) distributions on {0,1,2,…}, such that Γ₀ is the set of all compound-geometric distributions and Γ₁ the set of all compound-Poisson distributions, i.e. the set of all i.d. distributions on the non-negative integers. These classes are defined by recursion relations similar to those introduced by Katti [4] for Γ₁ and by Steutel [7] for Γ₀. These relations can be regarded as generalizations of those defining the so-called renewal sequences (cf. [5] and [2]). Several properties of i.d. distributions now appear as special cases of properties of the Γ_a'.     

The convolution of functions and distributions

The non-commutative convolution f∗g of two distributions f and g in D^′ is defined to be the limit of the sequence {(fτn)∗g}, provided the limit exists, where {τn} is a certain sequence of functions in D converging to 1. It is proved that

     

Infinitely divisible states, 1-cocycles, and conditionally positive functionals on algebras

Pairs of algebras in duality whose state spaces are semigroups (the duality in the sense of Vershik) are considered. A class of such algebras, which are called equipped, is introduced. For these algebras, the cone of conditionally positive functionals and its relationship with the geometry of the dual object and the 1-cocycles on with values in the *-representations are studied. In the case of group algebras, a symmetric construction describing infinitely divisible measures and states is given. Bibliography: 10 titles. Dedicated to L. D. Faddeev on the occasion of his 60th birthday Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 215, 1994, pp. 146–162. Translated by B. M. Bekker.     

Characterization of palm distributions and infinitely divisible random measures

The purpose of the present paper is to characterize Campbell measures and Palm distributions of random measures and to apply these results in a new approach to the characterizations of infinitely divisible random measures by their Laplace functionals and their Palm distributions. The results on infinitely divisible random measures are well known. They can be found together with a detailed list of references in Kallenberg's monograph [2], which also contains proofs of almost all statements in Section 1 of this paper (see his note on page 9 concerning the Polish space setting).     

Symmetry and classification of certain regular group divisible designs

In this paper it is shown that a regular group divisible (GD) design, with parametersv, b, r, k, ₁, ₂ satisfyingrk – ₂ v + 1 and ₂ = ₁ + 1, must be symmetric (i.e.,v + b). Furthermore, the parameters of such symmetric regular GD designs can be expressed in terms of only two integral parameters.Supported in part by Grant 59540043 (C), Japan.