Article

Annals of Combinatorics

, Volume 10, Issue 3, pp 333-341

Total Variation Distance for Poisson Subset Numbers

  • Larry GoldsteinAffiliated withDepartment of Mathematics, University of Southern California Email author 
  • , Gesine ReinertAffiliated withDepartment of Statistics, University of Oxford

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Abstract

Let n be an integer and A 0,..., A k random subsets of {1,..., n} of fixed sizes a 0,..., a k , respectively chosen independently and uniformly. We provide an explicit and easily computable total variation bound between the distance from the random variable \( W = {\left| { \cap ^{k}_{{j = 0}} A_{j} } \right|} \), 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 \( a_{j} \to \infty \) for all j = 0,..., k, showing that W has an asymptotic Poisson distribution in this regime.

AMS Subject Classification.

60C05 62E17

Keywords.

Poisson approximation Stein’s method size biasing surprisology