On generalized Poisson distribution on groups

G. M. Feldman

Teoriâ veroâtnostej i ee primeneniâ, Tome 20 (1975) no. 3, pp. 653-656

Résumé

The paper deals with the arithmetic of distributions on groups. Let $X$ be a locally compact Abelian separable metric group, $e(F)=e^{-F(X)}\biggl(E_0+F+\frac{F^{*2}}{2!}+\dots\biggl)$ is the generalized Poisson distribution associated with a finite measure $F$, and $I_0$ is a class of distributions without indecomposable or idempotent divisors. Some results are obtained on the conditions for generalized Poisson distributions to belong or not to belong to the class $I_0$. The density (in the weak topology) of the class $I_0$ in the set of all infinitely divisible distributions is also studied. If there is an element of the infinite order in any neighbourhood of zero in the group $X$, then the class $I_0$ is shown to be dense in the set of all infinitely divisible distributions. It is also proved that for discrete groups the density takes place if and only if $X\approx Z_2$.