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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
##### Keywords:
approximation problems; operator semigroup
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