Poisson Approximation for Unbounded Functions, I: Independent Summands

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 $\lambda$ 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 Universitat Z urich 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 maxfpni : 1 i ng ! 0 as n ! 1: Let Wn = P 1 k n Xnk and Z be a Poisson random variable with mean = 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 unrestricted. One of the theorems is then applied to obtain a large deviation result concerning Eh(Wn)I(Wn z) for a general class of functions h and again with unrestricted. The theorem is also applied to obtain an asymptotic result concerning P1 r=0 h((r )= p )jP (Wn = r) P (Z = r)j for large .