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^[1][2].

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

where ${\displaystyle E_{0}}$ is the degenerate distribution concentrated at the zero of the group ${\displaystyle X}$.

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

Decompositions of generalized Poisson distributions were studied in ^[3] and ^[4], see also ^[5]. In particular, the following theorem holds.

Theorem^[4]. Let ${\displaystyle X}$ be a locally compact Abelian group, and let ${\displaystyle F}$ be a finite non-negative measure on ${\displaystyle X}$. If the measures ${\displaystyle F^{*m}}$ and ${\displaystyle F^{*n}}$ are mutually singular for all natural ${\displaystyle m\neq n}$, then all factors of the generalized Poisson distribution ${\displaystyle \mu =e(F)}$ are also generalized Poisson distributions, i.e., ${\displaystyle \mu }$ has no indecomposable factors.