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Poisson approximation in a Poisson limit theorem inspired by coupon collecting. (English) Zbl 1173.60008

This paper is concerned with a Poisson approximation for the distribution of sums of asymptotically negligible integer-valued random variables in the setting of row-wise independent triangular arrays. The author refines the classical Poisson convergence theorem of B. V. Gnedenko and A. N. Kolmogorov [Limit distributions for sums of independent random variables. Translated from the Russian by K. L. Chung. Cambridge: Addison-Wesley Publishing Company (1954; Zbl 0056.36001)]. The upper and lower error bounds are expressed in terms of the total variation distance.

The results are applied to the case of the coupon collector’s problem when the distribution of the collector’s waiting time is asymptotically Poisson.

MSC:
60F05Central limit and other weak theorems
62E17Approximations to statistical distributions (nonasymptotic)
62E20Asymptotic distribution theory in statistics