Note on Pairwise Negative Dependence of Randomized Rank-1 Lattices

Abstract

In her recent paper [Negative dependence, scrambled nets, and variance bounds. Math. Oper. Res. 43 (2018), 228-251] Christiane Lemieux studied a framework to analyze the dependence structure of sampling schemes. The main goal of the framework is to determine conditions under which the negative dependence structure of a sampling scheme yields estimators with reduced variance compared to Monte Carlo estimators. For instance, she was able to show that in dimension d = 2 scrambled (0, m, d)-nets lead to randomized quasi-Monte Carlo estimators with variance no larger than the variance of Monte Carlo estimators for functions monotone in each variable. Her result relies on a pairwise negative dependence property that is, in particular, satisfied by (0, m, 2)-nets. In this note we establish that the same result holds true in arbitrary dimension d for a type of randomized lattice point sets that we call randomly shifted and jittered rank-1 lattices. We show that the details of the randomization are crucial and that already small modifications may destroy the pairwise negative dependence property.

Discrepancy Bounds for a Class of Negatively Dependent Random Points Including Latin Hypercube Samples

“…The proof techniques used there are based on a pairwise negative dependence condition and Hoeffding's lemma, see [42]. Further results in this direction are provided in [62,63,64].…”

A Generalized Faulhaber Inequality, Improved Bracketing Covers, and Applications to Discrepancy

“…,[CP76],[Owe95] or[l'Ec17]. A very recent research direction is to use notions of negative dependence to analyze random points sets, see e.g [GH19],[Lem18],[WG19],[WGH19],[WLD19]…”