Statistica Sinica

A. D. Barbour^{*},
Louis H. Y. Chen and K. P. Choi

Abstract:LetX,n≥1 be independent random variables with_{n1,}...,X_{nn}P(X=1)=1—_{ni}P(X=0)=_{ni}psuch that max{_{ni}p:1≤_{ni}i≤n} →0 asn→∞ LetW_{n}=Σ_{1≤k≤n}X_{nk}andzbe a Poisson random variable with meanλ=EWn. We obtain an absolute constant bound onP(W_{n}=r)/P(Z=r),r=0,1,... and using this, prove two Poisson approximation theorems forEH(W_{n}) withhunbounded andλunrestricted. One of the theorems is then applied to obtain a large deviation result concerningEH(W_{n})I(W_{n}≥z) for a general class of functionshand again withλunrestricted. The theorem is also applied to obtain an asymptotic result concerning for largeλ.

Key words and phrases:Poisson approximation, unbounded functions, large deviations, asymptotics, Stein's method.