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arXiv 1802.05070math.PR
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关于质点的拟无穷可分分布

On Quasi-Infinitely Divisible Distributions with a Point Mass

David Berger

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摘要：

$\mathbb{R}$上的无限可分分布是概率测度$μ$，使得特征函数$\hatμ$具有特征三元组$(a,γ, ν)$的Lévy-Khintchine表示，其中$ν$是Lévy测度，$γ\in\mathbb{R}$和$a\ge 0$。这种分布的自然延伸是拟无限分布。而不是一个Lévy措施，我们假设$ν$是一个“签署的Lévy措施”，为进一步的信息定义见[ \ref{Lindner} ]。我们证明了一个具有$p>0$和$x_0 \in \mathbb{R}$的分布$μ=pδ_{x_0}+(1-p)μ_{ac}$是拟无限可分的当且仅当$\hatμ(z)\neq0$对每个$z\in\mathbb{R}$都是绝对连续的.我们应用这一点证明了均值为零的正态分布的某些方差混合是拟无穷可分分布，并给出了一个不连续但具有无限拟Lévy测度的拟无限可分分布的例子。此外，它表明，取代签署的Lévy测度一个看似更一般的复杂Lévy测度不会导致新的分布。最后证明了拟无穷可分分布在概率测度空间中不是开的，而是具有Prokhorov度量的路径连通的。

英文摘要：

An infinitely divisible distribution on $\mathbb{R}$ is a probability measure $μ$ such that the characteristic function $\hatμ$ has a Lévy-Khintchine representation with characteristic triplet $(a,γ, ν)$, where $ν$ is a Lévy measure, $γ\in\mathbb{R}$ and $a\ge 0$. A natural extension of such distributions are quasi-infinitely distributions. Instead of a Lévy measure, we assume that $ν$ is a "signed Lévy measure", for further information on the definition see [\ref{Lindner}]. We show that a distribution $μ=pδ_{x_0}+(1-p)μ_{ac}$ with $p>0$ and $x_0 \in \mathbb{R}$, where $μ_{ac}$ is the absolutely continuous part, is quasi-infinitely divisible if and only if $\hatμ(z)\neq0$ for every $z\in\mathbb{R}$. We apply this to show that certain variance mixtures of mean zero normal distributions are quasi-infinitely divisible distributions, and we give an example of a quasi-infinitely divisible distribution that is not continuous but has infinite quasi-Lévy measure. Furthermore, it is shown that replacing the signed Lévy measure by a seemingly more general complex Lévy measure does not lead to new distributions. Last but not least it is proven that the class of quasi-infinitely divisible distributions is not open, but path-connected in the space of probability measures with the Prokhorov metric.

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