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Local and tail approximations near the Poisson limit. (English) Zbl 0674.60022

The authors improve earlier results of the first author [Ann. Probab. 15, 748-766 (1987; Zbl 0622.60049)] concerning the Poisson approximation of E h(W) and \(P(W=m)\) for a sum W of independent Bernoulli random variables. The bound for the error in the above mentioned paper may be large compared with \(P(W=m)\). The authors refine this bound in such a manner that the relative error is of order min(1,1/\(\sum^{N}_{i=1}p_ i)\sum^{N}_{i=1}p^ 1_ i\) if m belongs to the body of the distribution, i.e. \(| m-\lambda | \leq c\lambda^{1/2}.\)
Using the technique of conjugate distributions for W, tail approximations are obtained. The authors generalize these results to sums of non- negative integer valued random variables in the third part of the paper. A numerical example for the relative error of the approximation to the distribution of the sum \(B(3,0.5)+B(10,0.1)\) is given in the last part where the approximations from theorems 1 and 4 and a saddlepoint approximation are used.
Reviewer: F.Liese

MSC:

60F05 Central limit and other weak theorems
60G50 Sums of independent random variables; random walks

Citations:

Zbl 0622.60049
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