Generalized Poisson distribution on a locally compact Abelian group

Generalized Poisson distribution on a locally compact Abelian group, along with the Gaussian distribution, plays an important role in the arithmetic of probability distributions.

Let $X$ be a locally compact Abelian group, let $Y$ be its character group, and let $(x,y)$ be the value of a character $y\in Y$ at an element $x\in X$. Let $F$ be a finite non-negative measure on $X$. The generalized Poisson distribution associated with the measure $F$ is defined as a shift of the distribution $e(F)$ of the form

$e(F)=\exp \{-F(X)\}(E_0+F+\frac{F^{\ast 2}}{2!}+\dots +\frac{F^{\ast n}}{n!}+\dots ),$

where $E_0$ is the degenerate distribution concentrated at the zero of the group $X$.

The distribution $e(F)$ is infinitely divisible. The characteristic function of the distribution $e(F)$ has the form

$\widehat{e(F)}(y)=\exp \{{\int }_X[(x,y)-1]dF(x)\}.$

Decompositions of generalized Poisson distributions were studied in and , see also . In particular, the following theorem holds.

Theorem. Let $X$ be a locally compact Abelian group, and let $F$ be a finite non-negative measure on $X$. If the measures $F^{\ast m}$ and $F^{\ast n}$ are mutually singular for all natural $m\ne n$, then all factors of the generalized Poisson distribution $\mu =e(F)$ are also generalized Poisson distributions, i.e., $\mu$ has no indecomposable factors.