Poisson approximation for unbounded functions. I. Independent summands

Permanent URL to this publication: http://dx.doi.org/10.5167/uzh-22569

Barbour, A D; Chen, L; Choi, K (1995). Poisson approximation for unbounded functions. I. Independent summands. Statistica Sinica, 5(2):749-766.

Preview

PDF
1241Kb

Abstract

Let $X_{n1},\cdots,X_{nn},\ n\geq1$ , be independent random variables with $P(X_{ni}=1)=1-P(X_{ni}=0)=p_{ni}$ such that $\max\{p_{ni}\colon1\leq i\leq n\}\to0$ as $n\to\infty$ . Let $W_n=\sum_{1\leq k\leq n}X_{nk}$ and let $Z$ be a Poisson random variable with mean $\lambda=EW_n$ . We obtain an absolute constant bound on $P(W_n=r)/P(Z=r),\ r=0,1,\cdots$ , and using this, prove two Poisson approximation theorems for $Eh(W_n)$ with $h$ unbounded and $\lambda$ unrestricted. One of the theorems is then applied to obtain a large deviation result concerning $Eh(W_n)I (W_n\geq z)$ for a general class of functions $h$ and again with $\lambda$ unrestricted. The theorem is also applied to obtain an asymptotic result concerning $$$\sum$ ^ $\infty$ _ ${r=0}$ h((r- $\lambda$ )/ $\sqrt{\lambda}$ )|P(W_n=r)-P(Z=r)| $$$$ for large $\lambda$

Item Type:	Journal Article, refereed, original work
Communities & Collections:	07 Faculty of Science > Institute of Mathematics
DDC:	510 Mathematics
Uncontrolled Keywords:	Poisson approximation, unbounded functions, large deviations, asymptotics, Stein's method
Language:	English
Date:	1995
Deposited On:	09 Apr 2010 10:38
Last Modified:	23 Nov 2012 15:18
Publisher:	Academia Sinica, Institute of Statistical Science
ISSN:	1017-0405
Official URL:	http://www3.stat.sinica.edu.tw/statistica/j5n2/j5n223/j5n223.htm

Users (please log in): suggest update or correction for this item

Repository Staff Only: item control page