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About the density of spectral measure of the two-dimensional SaS random vector

Marta Borowiecka-OlszewskaJolanta K. Misiewicz — 2003

Discussiones Mathematicae Probability and Statistics

In this paper, we consider a symmetric α-stable p-sub-stable two-dimensional random vector. Our purpose is to show when the function e x p - ( | a | p + | b | p ) α / p is a characteristic function of such a vector for some p and α. The solution of this problem we can find in [3], in the language of isometric embeddings of Banach spaces. Our proof is based on simple properties of stable distributions and some characterization given in [4].

Simple fractions and linear decomposition of some convolutions of measures

Jolanta K. MisiewiczRoger Cooke — 2001

Discussiones Mathematicae Probability and Statistics

Every characteristic function φ can be written in the following way: φ(ξ) = 1/(h(ξ) + 1), where h(ξ) = ⎧ 1/φ(ξ) - 1 if φ(ξ) ≠ 0 ⎨ ⎩ ∞ if φ(ξ) = 0 This simple remark implies that every characteristic function can be treated as a simple fraction of the function h(ξ). In the paper, we consider a class C(φ) of all characteristic functions of the form φ a ( ξ ) = [ a / ( h ( ξ ) + a ) ] , where φ(ξ) is a fixed characteristic function. Using the well known theorem on simple fraction decomposition of rational functions we obtain that convolutions...

Classes of measures closed under mixing and convolution. Weak stability

For a random vector X with a fixed distribution μ we construct a class of distributions ℳ(μ) = μ∘λ: λ ∈ , which is the class of all distributions of random vectors XΘ, where Θ is independent of X and has distribution λ. The problem is to characterize the distributions μ for which ℳ(μ) is closed under convolution. This is equivalent to the characterization of the random vectors X such that for all random variables Θ₁, Θ₂ independent of X, X’ there exists a random variable Θ independent of X such...

Substable and pseudo-isotropic processes. Connections with the geometry of subspaces of L α

CONTENTSI. Introduction..........................................................................................................5II. 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....

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