Discrete Low-Discrepancy Sequences

Holroyd and Propp used Hall’s marriage theorem to show that, given a probability distribution π on a finite set S, there exists an infinite sequence s1, s2, . . . in S such that for all integers k ≥ 1 and all s in S, the number of i in [1, k] with si = s differs from k π(s) by at most 1. We prove a generalization of this result using a simple explicit algorithm. A special case of this algorithm yields an extension of Holroyd and Propp’s result to the case of discrete probability distributions on infinite sets.