Weak stability and generalized weak convolution for random vectors and stochastic processes

Abstract

A random vector $\mathbf{X}$ is weakly stable iff for all $a,b\in \mathbb{R}$ there exists a random variable $\Theta$ such that $a\mathbf{X}+b{\mathbf{X}}^{\prime }\stackrel{d}{=}\mathbf{X}\Theta$. 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 $$X{Q}_1+X^{\prime }{Q}_2\stackrel{d}{=}X\Theta ,$$ where $\mathbf{X},{\mathbf{X}}^{\prime },Q_1,Q_2,\Theta$ are independent. In this paper we define generalized convolution of measures defined by the formula $$L(Q_1){\oplus }_{\mu }L(Q_2)=L(\Theta ),$$ if the equation $(\ast )$ holds for $\mathbf{X},Q_1,Q_2,\Theta$ and $\mu =\mathcal{L}(\Theta )$. 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 at http://dx.doi.org/10.1214/074921706000000149 in the IMS Lecture Notes--Monograph Series (http://www.imstat.org/publications/lecnotes.htm) by the Institute of Mathematical Statistics (http://www.imstat.org)