# On Negatively Dependent Sampling Schemes, Variance Reduction, and Probabilistic Upper Discrepancy Bounds

## TL;DR

This paper explores negative dependence in sampling schemes to improve variance and discrepancy bounds, providing new explicit bounds and comparing different negative dependence notions.

## Contribution

It introduces new pre-asymptotic bounds with explicit constants for discrepancy measures under negative dependence conditions.

## Key findings

- Negative dependence can lead to improved variance bounds.
- Explicit constants are derived for star discrepancy bounds.
- Several negatively dependent sampling schemes are exemplified.

## Abstract

We study some notions of negative dependence of a sampling scheme that can be used to derive variance bounds for the corresponding estimator or discrepancy bounds for the underlying random point set that are at least as good as the corresponding bounds for plain Monte Carlo sampling. We provide new pre-asymptotic bounds with explicit constants for the star discrepancy and the weighted star discrepancy of sampling schemes that satisfy suitable negative dependence properties. Furthermore, we compare the different notions of negative dependence and give several examples of negatively dependent sampling schemes.