On the rate of Poisson convergence. (English) Zbl 0544.60029

Barbour, A. D.; Hall, Peter

Math. Proc. Camb. Philos. Soc. 95, 473-480 (1984).

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

Keywords:

Bernoulli random variables; total variation distance; Stein method; upper and lower bounds

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