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On the rate of Poisson convergence. (English) Zbl 0544.60029
Let $$X_ 1,...,X_ n$$ be independent Bernoulli random variables, and let $$p_ i=P[X_ i=1]$$, $$\lambda =\sum^{n}_{i=1}p_ i$$ and $$W=\sum^{n}_{i=1}X_ i$$. Denote by $$d({\mathcal L}(W),P_{\lambda})$$ the total variation distance between the distribution L(W) of W and a Poisson distribution $$P_{\lambda}$$ with mean $$\lambda$$. In this paper, using the Stein method, the authors obtain the following upper and lower bounds for $$d({\mathcal L}(W),P): (1/32)(1\wedge \lambda^{- 1})\sum^{n}_{j=1}p^ 2_ j\leq d({\mathcal L}(W),P_{\lambda})\leq \lambda^{-1}(1-e^{-\lambda})\sum^{n}_{j=1}p^ 2_ j.$$
Reviewer: H.Takahata

##### MSC:
 60F05 Central limit and other weak theorems 60E15 Inequalities; stochastic orderings
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##### References:
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