Probabilistic Star Discrepancy Bounds for Double Infinite Random Matrices

Abstract

In 2001 Heinrich, Novak, Wasilkowski and Woźniakowski proved that the inverse of the discrepancy depends linearly on the dimension, by showing that a Monte Carlo point set $\mathcal{P}$ of N points in the s-dimensional unit cube satisfies the discrepancy bound $D_{N}^{{\ast}s}(\mathcal{P}) \leq c_{\mathrm{abs}}{s}^{1/2}{N}^{-1/2}$ with positive probability. Later their results were generalized by Dick to the case of double infinite random matrices. In the present paper we give asymptotically optimal bounds for the discrepancy of such random matrices, and give estimates for the corresponding probabilities. In particular we prove that the N × s-dimensional projections $\mathcal{P}_{N,s}$ of a double infinite random matrix satisfy the discrepancy estimate

$$\displaystyle{D_{N}^{{\ast}s}(\mathcal{P}_{ N,s}) \leq {\left (2130 + 308\,\frac{\ln \ln N} {s} \right )}^{1/2}{s}^{1/2}{N}^{-1/2}}$$

for all N and s with positive probability. This improves the bound $D_{N}^{{\ast}s}(\mathcal{P}_{N,s}) \leq {\left (c_{\mathrm{abs}}\,\ln N\right )}^{1/2}{s}^{1/2}{N}^{-1/2}$ given by Dick. Additionally, we show how our approach can be used to show the existence of completely uniformly distributed sequences of small discrepancy which find applications in Markov Chain Monte Carlo.