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Classes of measures closed under mixing and convolution. Weak stability

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Misiewicz J. K., Oleszkiewicz K., Urbanik K.

Studia Mathematica

2005

tom 167

nr 3

195-213

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.

Ograniczanie wyników

1 Studia Mathematica

1 Misiewicz J. K.

1 Oleszkiewicz K.

1 Urbanik K.

1 2005

Wyniki wyszukiwania

Classes of measures closed under mixing and convolution. Weak stability