LIMIT THEOREMS FOR POWER-SERIES DISTRIBUTIONS WITH FINITE RADIUS OF CONVERGENCE.

Sufficient conditions for the weak convergence of the distributions of the random variables (1 - x)ξx as x → 1- to the limiting gamma-distribution are put forward. The random variable ξx has power-series distribution with radius of convergence 1 and parameter x ∈ (0, 1). Limit theorems for the probabilities P{ξx = k} are proposed. Asymptotic expansions of local probabilities are derived for sums of independent identically distributed variables with the same distribution as ξx in a triangular array with x → 1-. For the corresponding general allocation scheme, local limit theorems for the joint distributions of the occupancies of the cells are obtained. [ABSTRACT FROM AUTHOR]