Statistica Sinica 5(1995), 749-766

POISSON APPROXIMATION FOR UNBOUNDED

FUNCTIONS, I: INDEPENDENT SUMMANDS

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

Universität Zürich* and National University of Singapore

Abstract: Let Xn1,...,Xnn,n≥1 be independent random variables with P(Xni=1)=1—P(Xni=0)=pni such that max{pni:1≤in} →0 as n→∞ Let Wn1≤k≤nXnk and z be a Poisson random variable with mean λ=EWn . We obtain an absolute constant bound on P(Wn=r)/P(Z=r),r=0,1,... and using this, prove two Poisson approximation theorems for EH(Wn) with h unbounded and λ unrestricted. One of the theorems is then applied to obtain a large deviation result concerning EH(Wn)I(Wnz) for a general class of functions h and again with λ unrestricted. The theorem is also applied to obtain an asymptotic result concerning for large λ.

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