Poisson approximation for a sum of dependent indicators: An alternative approach

Abstract

The random variables X1, X2,...,Xn are said to be totally negatively dependent (TND) if and only if the random variables Xi and Σj≠i Xj are negatively quadrant dependent for all i. Our main result provides, for TND 0-1 indicators X1, X2,...,Xn with P[Xi = 1] = pi = 1 - P[Xi = 0], an upper bound for the total variation distance between Σi=1n Xi and a Poisson random variable with mean λ ≥ Σi=1n pi. An application to a generalized birthday problem is considered and, moreover, some related results concerning the existence of monotone couplings are discussed.