In 2001 Heinrich, Novak, Wasilkowski and Wo´zniakowski proved that for every s ≥ 1 and N ≥ 1 there exists a sequence (z 1 ,. .. , z N) of elements of the s-dimensional unit cube such that the star-discrepancy D * N of this sequence satisfies √ s √ N for some constant c independent of s and N. Their proof uses deep results from probability theory and… (More)
By a profound result of Heinrich, Novak, Wasilkowski, and Wo´zniakowski the inverse of the star-discrepancy n * (s, ε) satisfies the upper bound n * (s, ε) ≤ c abs sε −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 abs s 1/2 N −1/2. The proof is based on the… (More)
We prove the existence of a limit distribution for the normalized normality measure N (E N)/ √ N (as N → ∞) for random binary sequences E N , by this means confirming a conjecture of Alon, Kohayakawa, Mauduit, Moreira and Rödl. The key point of the proof is to approximate the distribution of the normality measure by the exiting probabilities of a… (More)
In many applications Monte Carlo (MC) sequences or Quasi-Monte Carlo (QMC) sequences are used for numerical integration. In moderate dimensions the QMC method typically yield better results, but its performance significantly falls off in quality if the dimension increases. One class of randomized QMC sequences, which try to combine the advantages of MC and… (More)
The problem of finding the largest empty axis-parallel box amidst a point configuration is a classical problem in computational complexity theory. It is known that the volume of the largest empty box is of asymptotic order 1/n for n → ∞ and fixed dimension d. However, it is natural to assume that the volume of the largest empty box increases as d gets… (More)
In the present paper we obtain fully explicit large deviation inequalities for empirical processes indexed by a Vapnik–Chervonenkis class of sets (or functions). Furthermore we illustrate the importance of such results for the theory of information-based complexity.
In a paper published in this journal, Alon, Kohayakawa, Mauduit, Moreira, and Rödl proved that the minimal possible value of the normality measure of an N-element binary sequence satisfies 1 2 + o(1)