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 X_n1,...,X_nn,n≥1 be independent random variables with P(X_ni=1)=1—P(X_ni=0)=p_ni such that max{p_ni:1≤i≤n} →0 as n→∞ Let W_n=Σ_1≤k≤nX_nk and z be a Poisson random variable with mean λ=EWn . We obtain an absolute constant bound on P(W_n=r)/P(Z=r),r=0,1,... and using this, prove two Poisson approximation theorems for EH(W_n) with h unbounded and λ unrestricted. One of the theorems is then applied to obtain a large deviation result concerning EH(W_n)I(W_n≥z) 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.