Total Variation Distance for Poisson Subset Numbers

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 = |∩ k 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 a asymptotic Poisson distribution in this regime.

Size bias for one and all

“…Implicit in Chen 1975 [25], with improved constants due to [15], see also [45, Theorem 4.12.12], is the following result from [42], Theorem 1.1, see also [77,Theorem 4.10], which we paraphrase 11 here as Theorem 5.1. Let X be a nonnegative integer valued random variable with λ := EX ∈ (0, ∞); let Z be Poisson with parameter λ.…”

“…To see how size bias enters, if a coupling satisfies P(X * ≤ X + c) = 1, then for all x, the event X * ≥ x is a subset of the event X ≥ x − c. Hence for x > 0, 11 The theorem in [42] is stated with the condition that X be a finite sum of indicator random variables. However, an arbitrary nonnegative integer valued X is a sum of indicators, namely X = i≥1 1(X ≥ i), and the restriction on finite sum can be removed using Theorem 2.3 applied to Xn := X ∧ n = n i=1 1(X ≥ i).…”

“…By the martingale property of X, for all i ≥ n ≥ 1, the righthand sides of (41) and (42) are equal to each other; hence (43) for i ≥ n ≥ 1, (X…”

Fixed points, descents, and inversions in parabolic double cosets of the symmetric group

“…We remark that Stein's method via size-bias coupling has been successfully used to prove Poisson limit theorems; some examples are: Angel, van der Hofstad, and Holmgren [1] for the number of self-loops and multiple edges in the configuration model, Arratia and DeSalvo [2] for completely effective error bounds on Stirling numbers of the first and second kinds, Goldstein and Reinert [19] for Poisson subset numbers, Holmgren and Janson for sums of functions of fringe subtrees of random binary search trees and random recursive trees, and Paguyo [34] for the number of simple chords in a random chord diagram.…”

Block Approximation of Tall Sparse Matrices and Block-Givens Rotations

“…The statement (a) is essentially borrowed from [6]. A zero-one vector u can be viewed as subset of a set its indexes.…”