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The random variables X 1, X 2, …, X n are said to be totally negatively dependent (TND) if and only if the random variables X i and ∑ j≠i 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 ∑ n i=1 X i and a Poisson random variable with mean λ ≥ ∑ n i=1 p i . An application to a generalized birthday problem is considered and, moreover, some related results concerning the existence of monotone couplings are discussed.
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