Some Asymptotic and Large Deviation Results in Poisson Approximation

Abstract

Let $X_{n1},\ldots, X_{nn}, n \geq 1$, be independent random variables with $P(X_{ni} = 1) = 1 - P(X_{ni} = 0) = p_{ni}$ such that $\max\{p_{ni}: 1 \leq i \leq n\}\rightarrow 0$ as $n\rightarrow\infty$. Let $W_n = \sum_{1\leq k\leq n} X_{nk}$ and let $Z$ be a Poisson random variable with mean $\lambda = EW_n$. Poisson approximation for the distribution of $W_n$ dates back to 1960, when Le Cam obtained upper bounds for the total variation distance $d(W_n, Z) = \sum_{k\geq 0}|P(W_n = k) - P(Z = k)|$. Barbour and Hall (in 1984) and Deheuvels and Pfeifer (in 1986) investigated the asymptotic behavior of $d(W_n, Z)$ as $n\rightarrow\infty$ for small, moderate and large $\lambda$. Their results imply that the orders of the bounds obtained by Le Cam are best possible. Chen proved (in 1974 and 1975) that the more general variation distance $d(h, W_n, Z) = \sum_{k\geq 0}h(k)|P(W_n = k) - P(Z = k)|, h \geq 0$, converges to 0 as $n\rightarrow\infty$, provided $\lambda$ remains bounded and $Eh(Z) < \infty$. We investigate the asymptotic behavior of: (i) $Eh(W_n) - Eh(Z_\lambda)$ for real $h$; and (ii) $d(h, W_n, Z)$ for $h \geq 0$ as $n\rightarrow\infty$ for small and moderate $\lambda$, thus generalizing the corresponding results of Barbour and Hall and of Deheuvels and Pfeifer. Our method also yields a large deviation result and holds promise for successful application in the case when $X_{n1},\ldots, X_{nn}$ are dependent.