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 λ ≥ ∑ni=1pi. An application to a generalized birthday problem is considered and, moreover, some related results concerning the existence of monotone couplings are discussed.
