Weak Lévy--Khintchine Representation for Weak Infinite Divisibility
| Autor: | J. K. Misiewicz, B. H. Jasiulis-Gołdyn |
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| Rok vydání: | 2016 |
| Předmět: | |
| Zdroj: | Theory of Probability & Its Applications. 60:45-61 |
| ISSN: | 1095-7219 0040-585X |
| DOI: | 10.1137/s0040585x97t987491 |
| Popis: | A random vector ${\bf X}$ is weakly stable if and only if 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 ${\bf X}'$ is an independent copy of ${\bf X}$ and $\Theta$ is independent of ${\bf X}$. This is equivalent (see [J. K. Misiewicz, K. Oleszkiewicz, and K. Urbanik, Studia Math., 167 (2005), pp. 195--213]) to the condition that for all random variables $Q_1, Q_2$ there exists a random variable $\Theta$ such that ${\bf X} Q_1 + {\bf X}' Q_2 \stackrel{d}{=} {\bf X} \Theta,$ where ${\bf X}, {\bf X}', Q_1, Q_2, \Theta$ are independent. In this paper we define a weak generalized convolution of measures defined by the formula ${\cal L}(Q_1) \otimes_{\mu} {\cal L}(Q_2) = {\cal L}(\Theta),$ if the former equation holds for ${\bf X}, Q_1, Q_2, \Theta$ and $\mu = {\cal L}(\bf X)$. We study here basic properties of this convolution and basic properties of distributions which are infinitely divisible in the sense of this ... |
| Databáze: | OpenAIRE |
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