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