This article is cited in 6 papers

Short Communications

Об абсолютной непрерывности безгранично делимых распределений при сдвигах

A. V. Skorokhod

Kiev

Abstract: Random variables $\xi$ with values in a separable Hilbert space $H$ with infinitely divisible distributions are considered. Some sufficient conditions for the absolute continuity of the measure corresponding to $\xi+a$ ($a\in H$) with respect to the measure corresponding to $\xi$ are obtained.
Let now $H$ denote the real line and let the characteristic function of $\xi$ be
$$ \exp\biggl\{\int\biggl(e^{ixt}-1-\frac{ixt}{1+x^2}\biggr)\Pi(dx)\biggr\}. $$
It is proved that in this case $\xi$ has a density when the condition $\int_{-1}^1|x|\Pi(dx)=\infty$ is satisfied.

Received: 14.01.1965

 English version: Theory of Probability and its Applications, 1965, 10:3, 465–472

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