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

TL;DR

This paper introduces a new class of negatively dependent random samples, including Latin hypercube samples, and provides probabilistic bounds on their star discrepancy, improving upon previous bounds for Monte Carlo and Latin hypercube sampling.

Contribution

The paper defines $oldsymbol{ extgamma}$-negatively dependent samples and derives optimal discrepancy bounds for this class, extending and improving existing results for Monte Carlo and Latin hypercube samples.

Findings

Provides probabilistic upper bounds for star discrepancy of $oldsymbol{ extgamma}$-negatively dependent samples.

Includes Latin hypercube samples within the new class and derives bounds for them.

Improves constants in discrepancy bounds for Monte Carlo samples.

Abstract

We introduce a class of $\gamma$-negatively dependent random samples. We prove that this class includes, apart from Monte Carlo samples, in particular Latin hypercube samples and Latin hypercube samples padded by Monte Carlo. For a $\gamma$-negatively dependent $N$-point sample in dimension $d$ we provide probabilistic upper bounds for its star discrepancy with explicitly stated dependence on $N$, $d$, and $\gamma$. These bounds generalize the probabilistic bounds for Monte Carlo samples from [Heinrich et al., Acta Arith. 96 (2001), 279--302] and [C.~Aistleitner, J.~Complexity 27 (2011), 531--540], and they are optimal for Monte Carlo and Latin hypercube samples. In the special case of Monte Carlo samples the constants that appear in our bounds improve substantially on the constants presented in the latter paper and in [C.~Aistleitner, M.~T.~Hofer, Math. Comp.~83 (2014), 1373--1381].