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

TL;DR

This paper introduces a new form of weak generalized convolution for measures based on weak stability, and establishes an analog of the Lévy-Khintchine theorem for distributions infinitely divisible under this convolution.

Contribution

It defines a novel weak generalized convolution for measures and proves a Lévy-Khintchine type representation for infinitely divisible distributions within this framework.

Findings

Defined weak generalized convolution of measures.

Characterized infinitely divisible distributions under this convolution.

Proved Lévy-Khintchine type representation theorem.

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.