Total Variation Distance for Poisson Subset Numbers

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 = | ∩j=0 A j|, 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 → ∞ for all j = 0, . . . , k, showing that W has an asymptotic Poisson distribution in this regime.