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 $${\bf X} Q_1 + {\bf X}' Q_2 \stackrel{d}{=} {\bf X} \Theta, %% \eqno{(\ast)}$$ 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.