×

zbMATH — the first resource for mathematics

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

MSC:
60F99 Limit theorems in probability theory
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Barbour, A. D. (1987). Asymptotic expansions in the Poisson limit theorem, Ann. Probab., 15, 748-766. · Zbl 0622.60049 · doi:10.1214/aop/1176992169
[2] Barbour, A. D. and Hall, P. (1984). On the rate of Poisson convergence, Math. Proc. Cambridge Philos. Soc., 95, 473-480. · Zbl 0544.60029 · doi:10.1017/S0305004100061806
[3] 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. · Zbl 0039.29002 · doi:10.1090/S0002-9904-1949-09223-6
[4] Deheuvels, P. and Pfeifer, D. (1986a). A semigroup approach to Poisson approximation, Ann. Probab., 14, 663-676. · Zbl 0597.60019 · doi:10.1214/aop/1176992536
[5] 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. · Zbl 0606.60034
[6] Deheuvels, P. and Pfeifer, D. (1987). Operator semigroups and Poisson convergence in selected metrics, Semigroup Forum, 34, 203-224. · Zbl 0606.60034 · doi:10.1007/BF02573163
[7] 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 Th?orique et Appliqu?e, Universit? Paris VI.
[8] LeCam, L. (1960). An approximation theorem for the Poisson binomial distribution, Pacific J. Math., 10, 1181-1197. · Zbl 0118.33601
[9] Pfeifer, D. (1983). A semi-group theoretic proof of Poisson’s limit law, Semigroup Forum, 26, 379-383. · Zbl 0507.60018 · doi:10.1007/BF02572846
[10] Pfeifer, D. (1985). A semigroup setting for distance measures in connexion with Poisson approximation, Semigroup Forum, 31, 201-205. · Zbl 0551.60039 · doi:10.1007/BF02572649
[11] Serfling, R. J. (1978). Some elementary results on Poisson approximation in a sequence of Bernoulli trials, SIAM Rev., 20, 567-579. · Zbl 0383.60027 · doi:10.1137/1020070
[12] Shorgin, S. Ya. (1977). Approximation of a generalized binomial distribution, Theory Probab. Appl., 22, 846-850. · Zbl 0392.60021 · doi:10.1137/1122099
[13] Uspensky, J. V. (1931). On Ch. Jordan’s series for probability, Ann. of Math., 32(2), 306-312. · Zbl 0002.20004 · doi:10.2307/1968193
[14] Zolotarev, V. M. (1984). Probability metrics, Theory Probab. Appl., 28, 278-302. · Zbl 0533.60025 · doi:10.1137/1128025
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.