Total Variation Distance for Poisson Subset Numbers

Abstract

Let n be an integer and A₀,..., A _k random subsets of {1,..., n} of fixed sizes a₀,..., 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.