On Negative Dependence Properties of Latin Hypercube Samples and Scrambled Nets

Abstract

We study the notion of ${\hat{I}}^3$-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 ${\hat{I}}^3$ that ensures ${\hat{I}}^3$-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. But it is known that the random variables based on Latin hypercube sampling in dimension $d$ are actually ${\hat{I}}^3$-negatively dependent with ${\hat{I}}^3\le e^d$, and the resulting probabilistic discrepancy bounds do only mildly depend on the ${\hat{I}}^3$-value.