New Bounds for the Extreme and the Star Discrepancy of Double-Infinite Matrices

Abstract

According to Aistleitner and Weimar, there exist two-dimensional (double) infinite matrices whose star-discrepancy $D_N^{\ast s}$ of the first $N$ rows and $s$ columns, interpreted as $N$ points in $[0,1{]}^s$, satisfies an inequality of the form $$D_N^{\ast s}\le \sqrt{\alpha }\sqrt{A+B\frac{\ln ({\log }_2(N))}{s}}\sqrt{\frac{s}{N}}$$ with $\alpha ={\zeta }^{-1}(2)\approx 1.73,A=1165$ and $B=178$. These matrices are obtained by using i.i.d sequences, and the parameters $s$ and $N$ refer to the dimension and the sample size respectively. In this paper, we improve their result in two directions: First, we change the character of the equation so that the constant $A$ gets replaced by a value $A_s$ dependent on the dimension $s$ such that for $s>1$ we have $A_s<A$. Second, we generalize the result to the case of the (extreme) discrepancy. The paper is complemented by a section where we show numerical results for the dependence of the parameter $A_s$ on $s$.

16 Pages

References in corpus (1)

• A note on a maximal Bernstein inequality