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

The random variables X1, X2, ., Xn are said to be totally negatively dependent (TND) if and only if the random variables Xi and .j.iXj 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 .ni=1Xi and a Poisson random variable with mean . . .ni=1pi. An application to a generalized birthday problem is considered and, moreover, some related results concerning the existence of monotone couplings are discussed.