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

The random variables X-1, X-2, X, are said to be totally negatively dependent (TND) if and only if the random variables X-i and Sigma(jnot equali) X-j are negatively quadrant dependent for all i. Our main result provides, for TND 0-1 indicators X-1, X-2,...,X-n with P[X-i = 1] = p(i) = 1-P[X-i = 0], an upper bound for the total variation distance between Sigma(i=1)(n) X-i and a Poisson random variable with mean; lambda greater than or equal to Sigma(i=1)(n) p(i). An application to a generalized birthday problem is considered and, moreover, some related results concerning the existence of monotone couplings are discussed.