Abstract
In this paper we show that Uspensky's expansion theorem 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 gives rise to exact evaluations and simple remainder term estimations for the deviations of the distributions in study with respect to various probability metrics, generalizing results of Shorgin (1977, Theory Probab. Appl., 22, 846–850). Finally, we compare the sharpness of Poisson versus normal approximations.
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References
Barbour, A. D. (1987). Asymptotic expansions in the Poisson limit theorem, Ann. Probab., 15, 748–766.
Barbour, A. D. and Hall, P. (1984). On the rate of Poisson convergence, Math. Proc. Cambridge Philos. Soc., 95, 473–480.
Cheng, B. (1964). The normal approximation to the Poisson distribution and a proot of a conjecture of Ramanujan, Bull. Amer. Math. Soc., 55, 396–401.
Deheuvels, P. and Pfeifer, D. (1986a). A semigroup approach to Poisson approximation, Ann. Probab., 14, 663–676.
Deheuvels, P. and Pfeifer, D. (1986b). Semigroups and Poisson approximation, Perspectives and New Directions in Theoretical and Applied Statistics, (eds. M. L., Puri, J. P., Villaplana and W., Wertz), Wiley, New York.
Deheuvels, P. and Pfeifer, D. (1987). Operator semigroups and Poisson convergence in selected metrics, Semigroup Forum, 34, 203–224.
Deheuvels, P., Puri, M. L. and Ralescu, S. S. (1986). Asymptotic expansions for sums of non identically distributed binomial random variables, Tech. Report, Laboratoire de Statistique Théorique et Appliquée, Université Paris VI.
LeCam, L. (1960). An approximation theorem for the Poisson binomial distribution, Pacific J. Math., 10, 1181–1197.
Pfeifer, D. (1983). A semi-group theoretic proof of Poisson's limit law, Semigroup Forum, 26, 379–383.
Pfeifer, D. (1985). A semigroup setting for distance measures in connexion with Poisson approximation, Semigroup Forum, 31, 201–205.
Serfling, R. J. (1978). Some elementary results on Poisson approximation in a sequence of Bernoulli trials, SIAM Rev., 20, 567–579.
Shorgin, S. Ya. (1977). Approximation of a generalized binomial distribution, Theory Probab. Appl., 22, 846–850.
Uspensky, J. V. (1931). On Ch. Jordan's series for probability, Ann. of Math., 32(2), 306–312.
Zolotarev, V. M. (1984). Probability metrics, Theory Probab. Appl., 28, 278–302.
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Deheuvels, P., Pfeifer, D. On a relationship between Uspensky's theorem and poisson approximations. Ann Inst Stat Math 40, 671–681 (1988). https://doi.org/10.1007/BF00049425
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DOI: https://doi.org/10.1007/BF00049425