Probabilistic discrepancy bound for Monte Carlo point sets

TL;DR

This paper extends a fundamental discrepancy bound for Monte Carlo point sets, providing explicit probabilistic guarantees and making the theoretical results more applicable for computational use.

Contribution

It establishes an explicit probabilistic bound on the star-discrepancy of random point sets, enhancing the theoretical result for practical computational applications.

Findings

Star-discrepancy of random sets is bounded with high probability.

Provides explicit constants for discrepancy bounds.

Applicable uniformly across different N and s.

Abstract

By a profound result of Heinrich, Novak, Wasilkowski, and Wo{\'z}niakowski the inverse of the star-discrepancy $n^*(s,\ve)$ satisfies the upper bound $n^*(s,\ve) \leq c_{\mathrm{abs}} s \ve^{-2}$. This is equivalent to the fact that for any $N$ and $s$ there exists a set of $N$ points in $[0,1]^s$ whose star-discrepancy is bounded by $c_{\mathrm{abs}} s^{1/2} N^{-1/2}$. The proof is based on the observation that a random point set satisfies the desired discrepancy bound with positive probability. In the present paper we prove an applied version of this result, making it applicable for computational purposes: for any given number $q \in (0,1)$ there exists an (explicitly stated) number $c(q)$ such that the star-discrepancy of a random set of $N$ points in $[0,1]^s$ is bounded by $c(q) s^{1/2} N^{-1/2}$ with probability at least $q$, uniformly in $N$ and $s$.