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 $s$.
Categories: math.NA math.PR