Probabilistic discrepancy bound for Monte Carlo point sets

Abstract:

By a profound result of Heinrich, Novak, Wasilkowski, and Woźniakowski the inverse of the star-discrepancy $n^*(s,\varepsilon )$ satisfies the upper bound $n^*(s,\varepsilon ) \leq c_{\mathrm {abs}} s \varepsilon ^{-2}$. 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 $s$.