Star discrepancy bounds based on Hilbert space filling curve stratified sampling and its applications

TL;DR

This paper introduces improved probabilistic star discrepancy bounds using Hilbert space filling curve stratified sampling, enhancing multivariate integral approximation and related inequalities with broader applicability.

Contribution

It presents a novel sampling method that improves star discrepancy bounds and convergence order, removing previous strict sampling conditions and extending applications.

Findings

Achieved better upper bounds for star discrepancy than Monte Carlo methods.

Improved convergence order from $O(N^{-1/2})$ to $O(N^{-1/2 - 1/(2d)})$.

Extended applications to Hilbert space sampling theorems and Koksma-Hlawka inequality.

Abstract

In this paper, we consider the upper bound of the probabilistic star discrepancy based on Hilbert space filling curve sampling. This problem originates from the multivariate integral approximation, but the main result removes the strict conditions on the sampling number of the classical grid-based jittered sampling. The main content has three parts. First, we inherit the advantages of this new sampling and achieve a better upper bound of the random star discrepancy than the use of Monte Carlo sampling. In addition, the convergence order of the upper bound is improved from $O(N^{-\frac{1}{2}})$ to $O(N^{-\frac{1}{2}-\frac{1}{2d}})$. Second, a better uniform integral approximation error bound of the function in the weighted space is obtained. Third, other applications will be given. Such as the sampling theorem in Hilbert spaces and the improvement of the classical Koksma-Hlawka inequality. Finally, the idea can also be applied to the proof of the strong partition principle of the star discrepancy version.