Open Access
VOL. 48 | 2006 Weak stability and generalized weak convolution for random vectors and stochastic processes
Jolanta K. Misiewicz

Editor(s) Dee Denteneer, Frank den Hollander, Evgeny Verbitskiy

IMS Lecture Notes Monogr. Ser., 2006: 109-118 (2006) DOI: 10.1214/074921706000000149


A random vector ${\bf X}$ is weakly stable iff for all $a,b \in \mathbb{R}$ there exists a random variable $\Theta$ such that $a{\bf X} + b {\bf X}' \stackrel{d}{=} {\bf X} \Theta$. This is equivalent (see \cite{MOU}) with the condition that for all random variables $Q_1, Q_2$ there exists a random variable $\Theta$ such that $${\bf X} Q_1 + {\bf X}' Q_2 \stackrel{d}{=} {\bf X} \Theta, %% \eqno{(\ast)}$$ where ${\bf X}, {\bf X}', Q_1, Q_2, \Theta$ are independent. In this paper we define generalized convolution of measures defined by the formula $${\cal L}(Q_1) \oplus_{\mu} {\cal L}(Q_2) = {\cal L}(\Theta),$$ if the equation $(\ast)$ holds for ${\bf X}, Q_1, Q_2, \Theta$ and $\mu = {\cal L}(\Theta)$. We study here basic properties of this convolution, basic properties of $\oplus_{\mu}$-infinitely divisible distributions, $\oplus_{\mu}$-stable distributions and give a series of examples.


Published: 1 January 2006
First available in Project Euclid: 28 November 2007

zbMATH: 1124.60003
MathSciNet: MR2306193

Digital Object Identifier: 10.1214/074921706000000149

Primary: 60A10 , 60B05 , 60E05 , 60E07 , 60E10

Keywords: scale mixture , symmetric stable distribution , weakly stable distribution

Rights: Copyright © 2006, Institute of Mathematical Statistics

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