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arXiv 1407.4097math.PR
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弱无穷可除性的弱Lévy-Khintchine表示

Weak Lévy-Khintchine representation for weak infinite divisibility

B. H. Jasiulis-Gołdyn, J. K. Misiewicz

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摘要：

随机向量${\bf X}$是弱稳定的当且仅当对所有的$a,b \in \mathbb{R}$都存在随机变量$Θ$使得$a{\bf X} + b {\bf X}' \stackrel{d}{=} {\bf X} Θ$，其中$X'$是$X$的独立拷贝，$Θ$独立于$X$。这等价于（参见[12]）对于所有随机变量$Q_1, Q_2$存在随机变量$Θ$使得$$ {\bf X} Q_1 + {\bf X}' Q_2 \stackrel{d}{=} {\bf X} Θ, \quad \quad \quad \quad \quad (\ast) $$其中${\bf X}, {\bf X}', Q_1, Q_2, Θ$是独立的条件。本文定义了由公式$$ {\mathcal L}(Q_1) \otimes_μ {\mathcal L}(Q_2) = {\mathcal L}(Θ), $$定义的测度的弱广义卷积，如果方程$(\ast)$对${\bf X}, Q_1, Q_2, Θ$和$μ= {\mathcal L}(X)$成立.我们在这里研究的基本性质的卷积和基本性质的分布是无限可分的意义上的卷积。本文的主要结果是$\otimes_μ$ -无穷可分分布的Lévy-Khintchine表示定理的类似结果。

英文摘要：

A random vector ${\bf X}$ is weakly stable iff for all $a,b \in \mathbb{R}$ there exists a random variable $Θ$ such that $a{\bf X} + b {\bf X}' \stackrel{d}{=} {\bf X} Θ$, where $X'$ is an independent copy of $X$ and $Θ$ 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 $Θ$ such that $$ {\bf X} Q_1 + {\bf X}' Q_2 \stackrel{d}{=} {\bf X} Θ, \quad \quad \quad \quad \quad (\ast) $$ where ${\bf X}, {\bf X}', Q_1, Q_2, Θ$ are independent. In this paper we define weak generalized convolution of measures defined by the formula $$ {\mathcal L}(Q_1) \otimes_μ {\mathcal L}(Q_2) = {\mathcal L}(Θ), $$ if the equation $(\ast)$ holds for ${\bf X}, Q_1, Q_2, Θ$ and $μ= {\mathcal L}(X)$. 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 Lévy-Khintchine representation theorem for $\otimes_μ$-infinitely divisible distributions.

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