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 distributionsK. R. Parthasarathy, R. Ranga Rao, V. S. Varadhan. Probability distributions on locally compact Abelian groups // Illinois Journal of Mathematics. — 1963. — Vol. 7. — P. 337–369. — https://doi.org/10.1215/ijm/1255644642.K. R. Parthasarathy. Probability measures on metric spaces. — New York; London: Academic Press, 1967. — (Probability and Mathematical Statistics; Vol. 3). — ..

Let be a locally compact Abelian group, let be its character group, and let be the value of a character at an element . Let be a finite non-negative measure on . The generalized Poisson distribution associated with the measure is defined as a shift of the distribution of the form

where is the degenerate distribution concentrated at the zero of the group .

The distribution is infinitely divisible. The characteristic function of the distribution has the form

Decompositions of generalized Poisson distributions were studied in G. M. Fel'dman. On a decomposition of generalized Poisson distributions on groups // Theory of Probability and Its Applications. — 1983. — Vol. 27, No. 4. — P. 780–794. — https://doi.org/10.1137/1127087. and G. M. Fel'dman. Generalized Poisson distributions of class I0 on groups // Theory of Probability and Its Applications. — 1985. — Vol. 29, No. 2. — P. 218–230. — https://doi.org/10.1137/1129034., see also G. M. Fel'dman. Arithmetic of Probability Distributions and Characterization Problems on Abelian Groups. Translations of Mathematical Monographs, Vol. 116. — Providence, RI: American Mathematical Society, 1993. — .. In particular, the following theorem holds.

Theorem. Let be a locally compact Abelian group, and let be a finite non-negative measure on . If the measures and are mutually singular for all natural , then all factors of the generalized Poisson distribution are also generalized Poisson distributions, i.e., has no indecomposable factors.

References