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On a relationship between Uspensky’s theorem and Poisson approximations. (English) Zbl 0675.60027
The authors show that a result of J. V. Uspensky [Ann. Math., II. Ser. 32, 306-312 (1931; Zbl 0002.20004)] for the Poisson approximation of the distribution of sums of independent Bernoulli random variables can be rewritten in terms of the Poisson convolution semigroup. This can be used in turn to derive bounds on the deviation of the distribution of sums of independent Bernoulli random variables from an appropriate Poisson distribution under various types of probability metrics. Results obtained generalize the work of S. Ya. Shorgin [Teor. Veroyatn. Primen. 22, 867-871 (1977; Zbl 0392.60021); English translation in Theory Probab. Appl. 22, 846-850 (1978)]. Sharpness of Poisson versus normal approximations is compared.
Reviewer: B.L.S.Prakasa Rao

60F99 Limit theorems in probability theory
Full Text: DOI
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