Permanent URL to this publication: http://dx.doi.org/10.5167/uzh22569
Barbour, A D; Chen, L; Choi, K (1995). Poisson approximation for unbounded functions. I. Independent summands. Statistica Sinica, 5(2):749766.

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Abstract
Let $X_{n1},\cdots,X_{nn},\ n\geq1$, be independent random variables with $P(X_{ni}=1)=1P(X_{ni}=0)=p_{ni}$ 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 $\lambda$
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Additional indexing
Item Type:  Journal Article, refereed, original work 

Communities & Collections:  07 Faculty of Science > Institute of Mathematics 
Dewey Decimal Classification:  510 Mathematics 
Uncontrolled Keywords:  Poisson approximation, unbounded functions, large deviations, asymptotics, Stein's method 
Language:  English 
Date:  1995 
Deposited On:  09 Apr 2010 08:38 
Last Modified:  05 Apr 2016 13:27 
Publisher:  Academia Sinica, Institute of Statistical Science 
ISSN:  10170405 
Official URL:  http://www3.stat.sinica.edu.tw/statistica/j5n2/j5n223/j5n223.htm 
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