Limit theorems for power-series distributions with finite radius of convergence

Sufficient conditions for the weak convergence of the distributions of the random variables $(1-x)\xi_x$ as $x\to1-$ to the limiting gamma-distribution are put forward. The random variable $\xi_x$ has power-series distribution with radius of convergence $1$ and parameter $x\in(0,1)$. Limit theorems for the probabilities $\mathbf P\{\xi_x=k\}$ are proposed. Asymptotic expansions of local probabilities are derived for sums of independent identically distributed variables with the same distribution as $\xi_x$ in a triangular array with $x\to1-$. For the corresponding general allocation scheme, local limit theorems for the joint distributions of the occupancies of the cells are obtained.