Geodesic
Limit theorems for power-series distributions with finite radius of convergence
Teoriâ veroâtnostej i ee primeneniâ, Tome 63 (2018) no. 1, pp. 57-69

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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.
Keywords: power-series distributions, radius of convergence, triangular arrays, weak convergence.
Mots-clés : gamma-distribution 
@article{TVP_2018_63_1_a2,
     author = {A. N. Timashev},
     title = {Limit theorems for power-series distributions with finite radius of convergence},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {57--69},
     publisher = {mathdoc},
     volume = {63},
     number = {1},
     year = {2018},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_2018_63_1_a2/}
}
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A. N. Timashev. Limit theorems for power-series distributions with finite radius of convergence. Teoriâ veroâtnostej i ee primeneniâ, Tome 63 (2018) no. 1, pp. 57-69. http://geodesic.mathdoc.fr/item/TVP_2018_63_1_a2/