Weak L\'evy-Khintchine representation for weak infinite divisibility

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 $a{\bf X} + b {\bf X}' \stackrel{d}{=} {\bf X} \Theta$, where $X'$ is an independent copy of $X$ and $\Theta$ is independent of $X$. This is equivalent (see [12]) with the condition that for all random variables $Q_1, Q_2$ there exists a random variable $\Theta$ such that $$ {\bf X} Q_1 + {\bf X}' Q_2 \stackrel{d}{=} {\bf X} \Theta, \quad \quad \quad \quad \quad (\ast) $$ where ${\bf X}, {\bf X}', Q_1, Q_2, \Theta$ are independent. In this paper we define weak generalized convolution of measures defined by the formula $$ {\mathcal L}(Q_1) \otimes_{\mu} {\mathcal L}(Q_2) = {\mathcal L}(\Theta), $$ if the equation $(\ast)$ holds for ${\bf X}, Q_1, Q_2, \Theta$ and $\mu = {\mathcal L}(X)$. We study here basic properties of this convolution and basic properties of distributions which are infinitely divisible in the sense of this convolution. The main result of this paper is the analog of the L\'evy-Khintchine representation theorem for $\otimes_{\mu}$-infinitely divisible distributions.