Annals of Combinatorics

, Volume 10, Issue 3, pp 333–341

Total Variation Distance for Poisson Subset Numbers

Authors

    • Department of MathematicsUniversity of Southern California
  • Gesine Reinert
    • Department of StatisticsUniversity of Oxford
Article

DOI: 10.1007/s00026-006-0291-9

Cite this article as:
Goldstein, L. & Reinert, G. Ann. Comb. (2006) 10: 333. doi:10.1007/s00026-006-0291-9
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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 \( W = {\left| { \cap ^{k}_{{j = 0}} A_{j} } \right|} \), 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.

60C0562E17

Keywords.

Poisson approximationStein’s methodsize biasingsurprisology

Copyright information

© Birkhäuser Verlag, Basel 2006