# Poisson approximation for unbounded functions. I. Independent summands

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

## 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 $\lambda$

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 $\lambda$

## Citations

Detailed statistics

Item Type: Journal Article, refereed, original work 07 Faculty of Science > Institute of Mathematics 510 Mathematics Poisson approximation, unbounded functions, large deviations, asymptotics, Stein's method English 1995 09 Apr 2010 08:38 05 Apr 2016 13:27 Academia Sinica, Institute of Statistical Science 1017-0405 http://www3.stat.sinica.edu.tw/statistica/j5n2/j5n223/j5n223.htm
Permanent URL: http://doi.org/10.5167/uzh-22569