On negative dependence properties of Latin hypercube samples and scrambled nets

Abstract

We study the notion of gamma-negative dependence of random variables. This notion is a relaxation of the notion of negative orthant dependence (which corresponds to 1-negative dependence), but nevertheless it still ensures concentration of measure and allows to use large deviation bounds of Chernoff-Hoeffding-or Bernstein type. We study random variables based on random points P. These random variables appear naturally in the analysis of the discrepancy of P or, equivalently, of a suitable worst-case integration error of the quasi-Monte Carlo cubature that uses the points in P as integration nodes. We introduce the correlation number, which is the smallest possible value of gamma that ensures gamma-negative dependence. We prove that the random variables of interest based on Latin hypercube sampling or on (t, m, d)-nets do, in general, not have a correlation number of 1, i.e., they are not negative orthant dependent. (C) 2021 Elsevier Inc. All rights reserved.