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

Abstract

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 $$X Q_1 + X' Q_2 \stackrel{d}{=} X \Theta,$$ 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 $$L(Q_1) \oplus_{\mu} L(Q_2) = L(\Theta),$$ if the equation $(*)$ 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.Comment: 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