Об абсолютной непрерывности безгранично делимых распределений при сдвигах
Teoriâ veroâtnostej i ee primeneniâ, Tome 10 (1965) no. 3, pp. 510-518
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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.
@article{TVP_1965_10_3_a8,
author = {A. V. Skorokhod},
title = {{\CYRO}{\cyrb} {\cyra}{\cyrb}{\cyrs}{\cyro}{\cyrl}{\cyryu}{\cyrt}{\cyrn}{\cyro}{\cyrishrt} {\cyrn}{\cyre}{\cyrp}{\cyrr}{\cyre}{\cyrr}{\cyrery}{\cyrv}{\cyrn}{\cyro}{\cyrs}{\cyrt}{\cyri} {\cyrb}{\cyre}{\cyrz}{\cyrg}{\cyrr}{\cyra}{\cyrn}{\cyri}{\cyrch}{\cyrn}{\cyro} {\cyrd}{\cyre}{\cyrl}{\cyri}{\cyrm}{\cyrery}{\cyrh} {\cyrr}{\cyra}{\cyrs}{\cyrp}{\cyrr}{\cyre}{\cyrd}{\cyre}{\cyrl}{\cyre}{\cyrn}{\cyri}{\cyrishrt} {\cyrp}{\cyrr}{\cyri} {\cyrs}{\cyrd}{\cyrv}{\cyri}{\cyrg}{\cyra}{\cyrh}},
journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
pages = {510--518},
publisher = {mathdoc},
volume = {10},
number = {3},
year = {1965},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/TVP_1965_10_3_a8/}
}
A. V. Skorokhod. Об абсолютной непрерывности безгранично делимых распределений при сдвигах. Teoriâ veroâtnostej i ee primeneniâ, Tome 10 (1965) no. 3, pp. 510-518. http://geodesic.mathdoc.fr/item/TVP_1965_10_3_a8/