The BK inequality for pivotal sampling a.k.a. the Srinivasan sampling process

Abstract

The pivotal sampling algorithm, a.k.a. the Srinivasan sampling process, is a simply described recursive algorithm for sampling from a finite population a fixed number of items such that each item is included in the sample with a prescribed desired inclusion probability.The algorithm has attracted quite some interest in recent years due to the fact that despite its simplicity, it has been shown to satisfy strong properties of negative dependence, e.g. conditional negative association.In this paper it is shown that (tree-ordered) pivotal/Srinivasan sampling also satisfies the BK inequality.This is done via a mapping from increasing sets of samples to sets of match sequencesand an application of the van den Berg-Kesten-Reimer inequality.The result is one of only very few non-trivial situations where the BK inequality is known to hold.<br />