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Total Variation Distance for Poisson Subset Numbers
Larry Goldstein1 and Gesine Reinert2
1Department of Mathematics, University of Southern California, 3620 Vermont Avenue, Los Angeles, CA 90089-2532, USA
larry@math.usc.edu
2Department of Statistics, University of Oxford, 1 South Parks Road, Oxford OX1 3TG, UK
reinert@stats.ox.ac.uk
Annals of Combinatorics 10 (3) p. 333-341 September, 2006
AMS Subject Classification: 60C05, 62E17
Abstract:
Let n be an integer and A0, ... , Ak random subsets of {1,... , n} of fixed sizes a0, ... , ak, respectively chosen independently and uniformly. We provide an explicit and easily computable total variation bound between the distance from the random variable W = |∩kj=0 Aj|, the size of the intersection of the random sets, to a Poisson random variable Z with intensity λ = EW. In particular, the bound tends to zero when λ converges and aj→ ∞, for all j = 0, ... ,k, showing that W has an asymptotic Poisson distribution in this regime.
Keywords: Poisson approximation, Stein's method, size biasing, surprisology

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