Abstract
A generalized multivariate problem due to V. M. Zolotarev is considered. Some related results on geometric random sums and (multivariate) geometric stable distributions are extended to a more general case of “anisotropic” random summation where sums of independent random vectors with multivariate random index having a special multivariate geometric distribution are considered. Anisotropic-geometric stable distributions are introduced. It is demonstrated that these distributions are coordinate-wise scale mixtures of elliptically contoured stable distributions with the Marshall–Olkin mixing distributions. The corresponding “anisotropic” analogs of multivariate Laplace, Linnik and Mittag–Leffler distributions are introduced. Some relations between these distributions are presented.
Highlights
The symbol denotes the operation of coordinate-wise multiplication of independent random vectors
The set of admissible multivariate geometric distributions should be narrowed to some sub-family P. As it has already been mentioned, in many papers the case was considered where p1 = p ∈ (0, 1), p0 = 1 − p. This corresponds to the case where random sums of random vectors are considered with the univariate random index having the univariate geometric distribution
The general case of the multivariate summation index with the multivariate geometric distribution is for the first time considered in this paper
Summary
Assume that all the random variables and random vectors are defined on one and the same probability space (Ω, A, P). The characteristic function g(t) of a strictly stable random variable can be represented in several equivalent forms (see, e.g., [1,2]). By S(α, 1) we will denote a positive random variable with the one-sided stable distribution corresponding to the characteristic function gα,1(t), t ∈ R. Let μ be a finite (‘spectral’) measure on Qd. It is known that the characteristic function of a strictly stable random vector S has the form. A d-variate analog of a one-sided univariate strictly stable random variable S(α, 1) is the random vector S(α, μ+) where 0 < α ≤ 1 and μ+ is a finite measure concentrated on the set Qd+ = {u = Anisotropic multivariate Linnik and Mittag–Leffler distributions are introduced and some of their properties are discussed
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