On Negative Dependence Properties of Latin Hypercube Samples and Scrambled Nets
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We study the notion of γ-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 γ that ensures γ-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 γ-negatively dependent with γ≤ e^d, and the resulting probabilistic discrepancy bounds do only mildly depend on the γ-value.
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