Abstract
We establish numerous new refined local limit theorems for a class of com- pound Poisson processes with P´ olya-Aeppli marginals, and for a particular family of the branching particle systems which undergo critical binary branching and can be approxi- mated by the backshifted Feller diffusion. To this end, we also derive new results for the families of P´ olya-Aeppli and Poisson-exponential distributions. We relate a few of them to properties of certain special functions some of which were previously unknown. n ; n 1g is a sequence of geometrically distributed i.i.d. random vari- ables (or r.v.'s) whose range is N, and which are characterized by the probability of suc- cess 2 (0; 1). In addition, they are assumed to be independent of the Poisson counting process f � (t); t 0g with intensity � > 0. In view of (9), (26), (28), the marginals of both these classes of stochastic processes belong to the two-parameter P´ olya-Aeppli fam-
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