SOME ASYMPTOTIC AND LARGE DEVIATION RESULTS IN POISSON APPROXIMATION

Let X(n1),...,X(nn), n greater-than-or-equal-to 1, be independent random variables with P(X(ni) = 1) = 1 - P(X(ni) = 0) = p(ni) such that max{p(ni): 1 less-than-or-equal-to i less-than-or-equal-to n} --> 0 as n --> infinity. Let W(n) = SIGMA1 less-than-or-equal-to k less-than-or-equal-to 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) = SIGMA(k greater-than-or-equal-to 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 --> infinity 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) = SIGMA(k greater-than-or-equal-to 0)h(k)\P(W(n) = k) - P(Z = k)\, h greater-than-or-equal-to 0, converges to 0 as n --> infinity, provided lambda remains bounded and Eh(Z) < infinity. 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 greater-than-or-equal-to 0 as n --> infinity 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),...,X(nn) are dependent.