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Short Communications

On a decomposition of a Gaussian distribution on groups

G. M. Fel'dman

Institute for Low Temperature Physics and Engineering, Academy of Sciences of Ukrainian SSR, Har'kov

Abstract: Let $X$ be a connected locally compact Abelian separable metric group.
The following generalization of Cramer's theorem is obtained: an arbitrary Gaussian distribution $\mu$ on the group $X$ has only Gaussian divisors if and only if $X$ does not contain a subgroup isomorphic to the circle group T.
It is also shown that any Gaussian distribution $\mu$, the support of which coincides with $X$, has a non-Gaussian divisor if and only if the group $X$ is isomorphic to a group of the form $R^p\times T$, $p\ge 0$.

Received: 30.09.1975

 English version: Theory of Probability and its Applications, 1977, 22:1, 133–140

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