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A semigroup approach to Poisson approximation. (English) Zbl 0597.60019
Let \(X_ 1,...,X_ n\) be independent Bernoulli r.v.’s with \(p_ j=P(X_ j=1)=1-P(X_ j=0)\), \(0<p_ j<1\), \(j=1,2,...,n\) and \(Y_ 1,...,Y_ n\) be independent Poisson random variables with expectations \(\mu_ j\), \(j=1,2,...,n\) and define the following sums \(S_ n=\sum^{n}_{j=1}X_ j\), \(T_ n=\sum^{n}_{j=1}Y_ j\). The authors investigate the variation distance \[ d(S_ n,T)=\sup_{M\subset {\mathbb{Z}}^+}| P(S_ n\in M)-P(T_ n\in M)| \] and the aim of this paper is: (1) to show that Poisson approximation problems for independent summands can be treated in a suitable operator semigroup framework and (2) to determine asymptotically those Poisson distributions which minimize this distance for given Bernoulli summands.
Reviewer’s remark: Unfortunately, the authors are unaware of a classical work due to Yu. V. Prokhorov [Usp. Mat. Nauk 8, No.3(55), 135-142 (1953; Zbl 0051.103)]. A comparison of Prokhorov’s work and the work under review would be of interest. For more recent results along the same line see: E. Presman, Teor. Veroyatn. Primen. 30, No.2, 391-396 (1985; Zbl 0568.60026).
Reviewer: N.Gamkrelidze

MSC:
60F05 Central limit and other weak theorems
62E20 Asymptotic distribution theory in statistics
47D03 Groups and semigroups of linear operators
20M30 Representation of semigroups; actions of semigroups on sets
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