Speed and concentration of the covering time for structured coupon collectors

Abstract

Let V be an n-set, and let X be a random variable taking values in the powerset of V . Suppose we are given a sequence of random coupons X 1 , X 2 , . . ., where the X i are independent random variables with distribution given by X. The covering time T is the smallest integer t ≥ 0 such that t i=1 X i = V . The distribution of T is important in many applications in combinatorial probability, and has been extensively studied. However the literature has focussed almost exclusively on the case where X is assumed to be symmetric and/or uniform in some way.In this paper we study the covering time for much more general random variables X; we give general criteria for T being sharply concentrated around its mean, precise tools to estimate that mean, as well as examples where T fails to be concentrated and when structural properties in the distribution of X allow for a very different behaviour of T relative to the symmetric/uniform case.

Connectivity of Poissonian inhomogeneous random multigraphs

“…We will prove that the existence of isolated points in G n (P) is regulated mainly by the first moment of their number. This result can be deduced from the main theorem of [7] with some effort, seeing the edge addition process as a coupon collector over the vertices, but we provide here a short and direct proof to improve readability of the paper.…”

Biased random k-SAT

“…It is worth noting that random SAT problems are examples of structured coupon collector problems [13]. Each assignment of true/false to the n Boolean variables is an n-word (ε 1 , ε 2 , .…”