Classes of measures closed under mixing and convolution. Weak stability

Jolanta K. Misiewicz, Krzysztof Oleszkiewicz, and Kazimierz Urbanik. "Classes of measures closed under mixing and convolution. Weak stability." Studia Mathematica 167.3 (2005): 195-213. <http://eudml.org/doc/284446>.

@article{JolantaK2005,
abstract = {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 that $XΘ₁ + X^\{\prime \}Θ₂ \stackrel\{d\}\{=\} XΘ$. We show that for every X this property is equivalent to the following condition: ∀ a,b ∈ ℝ ∃ Θ independent of X, $aX + bX^\{\prime \} \stackrel\{d\}\{=\} XΘ$. This condition reminds the characterizing condition for symmetric stable random vectors, except that Θ is here a random variable, instead of a constant. The above problem has a direct connection with the concept of generalized convolutions and with the characterization of the extreme points for the set of pseudo-isotropic distributions.},
author = {Jolanta K. Misiewicz, Krzysztof Oleszkiewicz, Kazimierz Urbanik},
journal = {Studia Mathematica},
keywords = {convolution; generalized convolution; pseudo-isotropic distributions; elliptically contoured distributions; weakly stable measures},
language = {eng},
number = {3},
pages = {195-213},
title = {Classes of measures closed under mixing and convolution. Weak stability},
url = {http://eudml.org/doc/284446},
volume = {167},
year = {2005},
}

TY - JOUR
AU - Jolanta K. Misiewicz
AU - Krzysztof Oleszkiewicz
AU - Kazimierz Urbanik
TI - Classes of measures closed under mixing and convolution. Weak stability
JO - Studia Mathematica
PY - 2005
VL - 167
IS - 3
SP - 195
EP - 213
AB - 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 that $XΘ₁ + X^{\prime }Θ₂ \stackrel{d}{=} XΘ$. We show that for every X this property is equivalent to the following condition: ∀ a,b ∈ ℝ ∃ Θ independent of X, $aX + bX^{\prime } \stackrel{d}{=} XΘ$. This condition reminds the characterizing condition for symmetric stable random vectors, except that Θ is here a random variable, instead of a constant. The above problem has a direct connection with the concept of generalized convolutions and with the characterization of the extreme points for the set of pseudo-isotropic distributions.
LA - eng
KW - convolution; generalized convolution; pseudo-isotropic distributions; elliptically contoured distributions; weakly stable measures
UR - http://eudml.org/doc/284446
ER -