Abstract
We present a unified and self-contained approach to Poisson approximation problems for independent Bernoulli summands with respect to several metrics by a general semigroup technique, expanding and completing earlier work on this subject by the first two authors [4], [5], [6].
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References
Barbour, A. D. (1987),Asymptotic expansions in the Poisson limit theorem, Ann. Prob.15, 748–766.
Barbour, A. D. and Hall, P. (1984),On the rate of Poisson convergence, Math. Proc. Cambridge Philos. Soc.95, 473–480.
Bruss, F. T. (1984),Patterns of relative maxima in random permutations, Ann. Soc. Sci. Bruxelles98 (I), 19–28.
Deheuvels, P. and Pfeifer, D. (1986a),A semigroup approach to Poisson approximation, Ann. Prob.14, 663–676.
Deheuvels, P. and Pfeifer, D. (1986b),Operator semigroups and Poisson convergence in selected metrics, Semigroup Forum34, 203–224.
Deheuvels, P. and Pfeifer, D. (1987),Semigroups and Poisson approximation.In:New Perspectives in Theoretical and Applied Statistics, M. L. Puri, J. P. Vilaplana, W. Wertz (eds.), John Wiley, N.Y., 439–448.
Deheuvels, P., Karr, A. F., Pfeifer, D. and Serfling, R. J. (1988),Poisson approximations in selected metrics by coupling and semigroup methods with applications. To appear in: J. Stat. Planning Inf.
Gastwirth, J. L. and Bhattacharya, P. K. (1984),Probability models of pyramid or chain letter systems demonstrating that their promotional claims are unreliable, Operat. Res.32, 527–536.
Hipp, C. (1985),Approximation of aggregate claims distributions by compound Poisson distributions, Insurance: Math. and Econ.4, 227–232.
Johnson, N. L. and Simons, G. (1971),On the convergence of binomial to Poisson distributions, Ann. Math. Stat.42, 1735–1736.
Kemp, R. (1984),Fundamentals of the Average Case Analysis of Particular Algorithms, Wiley-Teubner, N.Y.
Le Cam, L. (1960),An approximation theorem for the Poisson binomial distribution, Pacific J. Math.10, 1181–1197.
Nevzorov, V. B. (1986),Two characterizations using records.In:Stability Problems for Stochastic Models, Lecture Notes in Math.1233, Springer, Berlin, 79–85.
Pfeifer, D. (1986),Extremal processes, record times and strong approximation, Pub. Inst. Stat. Univ. Paris31, 47–65.
Pfeifer, D. (1988),Extremal processes, secretary problems and the 1/e-law. To appear in: J. Appl. Prob.
Rényi, A. (1962),Théorie des éléments saillants d’une suite d’observations, Coll. Comb. Meth. Prob. Th.104–115, Mathematisk Institut, Aarhus Universitet, Denmark.
Ross, S. M. (1982),A simple heuristic approach to simplex efficiency, Eur. J. Oper. Res.9, 344–346.
Serfling, R. J. (1975),A general Poisson approximation theorem, Ann. Prob.3, 726–731.
Serfling, R. J. (1978),Some elementary results on Poisson approximation in a sequence of Bernoulli trials, SIAM Rev.20, 567–579.
Shorgin, S. Y. (1977),Approximation of a generalized binomial distribution, Th. Prob. Appl.22, 846–850.
Zolotarev, V. M. (1976),Metric distances in spaces of random variables and their distributions, Math. U.S.S.R. Sbornik30, 3, 373–401.
Zolotarev, V. M. (1984),Probability metrics, Th. Prob. Appl.28, 278–302.
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Deheuvels, P., Pfeifer, D. & Puri, M.L. A new semigroup technique in poisson approximation. Semigroup Forum 38, 189–201 (1989). https://doi.org/10.1007/BF02573230
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DOI: https://doi.org/10.1007/BF02573230