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## Studia Mathematica

2005 | 167 | 3 | 195-213
Tytuł artykułu

### Classes of measures closed under mixing and convolution. Weak stability

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Abstrakty
EN
For a random vector X with a fixed distribution μ we construct a class of distributions ℳ(μ) = {μ∘λ: λ ∈ 𝓟}, 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
$XΘ₁ + X'Θ₂ \stackrel{d}{=} XΘ$.
We show that for every X this property is equivalent to the following condition:
∀ a,b ∈ ℝ ∃ Θ independent of X, $aX + bX' \stackrel{d}{=} XΘ$.
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.
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Tom
Numer
Strony
195-213
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Daty
wydano
2005
Twórcy
autor
• Department of Mathematics, Informatics and Econometry, University of Zielona Góra, Podgórna 50, 65-246 Zielona Góra, Poland
autor
• Institute of Mathematics, Warsaw University, Banacha 2, 02-097 Warszawa, Poland
autor
• Institute of Mathematics, University of Wrocław, Pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland
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