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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

60F05 Central limit and other weak theorems
60E15 Inequalities; stochastic orderings
Full Text: DOI
[1] DOI: 10.1111/j.1467-9574.1969.tb00075.x · Zbl 0162.22203 · doi:10.1111/j.1467-9574.1969.tb00075.x
[2] DOI: 10.1214/aop/1176996313 · Zbl 0321.60018 · doi:10.1214/aop/1176996313
[3] DOI: 10.1111/j.1467-9574.1977.tb00759.x · Zbl 0369.60025 · doi:10.1111/j.1467-9574.1977.tb00759.x
[4] DOI: 10.2307/1426620 · Zbl 0511.60025 · doi:10.2307/1426620
[5] Cam, Pacific J. Math. 10 pp 1181– (1960) · Zbl 0118.33601 · doi:10.2140/pjm.1960.10.1181
[6] DOI: 10.1007/BF00533378 · Zbl 0123.35403 · doi:10.1007/BF00533378
[7] DOI: 10.1214/aop/1176996359 · Zbl 0335.60016 · doi:10.1214/aop/1176996359
[8] Prohorov, Uspekhi Mat. Nauk 8 pp 135– (1953)
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