On the Distribution of Digital Sequences

Low-discrepancy point sets are the basic tool for various quasi-Monte Carlo methods. The most important concept for the construction of low-discrepancy point sets in an s-dimensional unit cube is the concept of digital point sets, introduced by Niederreiter in 1987. We show that “almost all” digital sequences in a prime base are, in a certain sense, almost best possible distributed in the unit cube. Thereby we improve a result given in [LN95]

[1]  H. Niederreiter,et al.  Digital nets and sequences constructed over finite rings and their application to quasi-Monte Carlo integration , 1996 .

[2]  Gerhard Larcher.  A bound for the discrepancy of digital nets and its application to the analysis of certain pseudo-random number generators , 1998 .

[3]  H. Niederreiter.  Low-discrepancy and low-dispersion sequences , 1988 .

[4]  Harald Niederreiter,et al.  Quasirandom points and global function fields , 1996 .

[5]  H. Niederreiter,et al.  Low-Discrepancy Sequences and Global Function Fields with Many Rational Places , 1996 .

[6]  H. Niederreiter.  Point sets and sequences with small discrepancy , 1987 .

[7]  Wolfgang Ch. Schmid,et al.  Multivariate Walsh series, digital nets and quasi-Monte Carlo integration , 1995 .

[8]  Harald Niederreiter,et al.  Low-discrepancy point sets obtained by digital constructions over finite fields , 1992 .

[9]  Harald Niederreiter,et al.  Optimal Polynomials for ( t,m,s )-Nets and Numerical Integration of Multivariate Walsh Series , 1996 .

[10]  H. Faure.  Discrépance de suites associées à un système de numération (en dimension s) , 1982 .

[11]  Harald Niederreiter,et al.  Random number generation and Quasi-Monte Carlo methods , 1992, CBMS-NSF regional conference series in applied mathematics.

[12]  Harald Niederreiter,et al.  Generalized $(t,s)$-sequences, Kronecker-type sequences, and Diophantine approximations of formal Laurent series , 1995 .

[13]  Gerhard Larcher.  On the Distribution of an Analog to Classical Kronecker-Sequences , 1995 .

[14]  Gerhard Larcher,et al.  On the numerical integration of Walsh series by number-theoretic methods , 1994 .

[15]  I. Sobol.  On the distribution of points in a cube and the approximate evaluation of integrals , 1967 .

[16]  K. F. Roth.  On irregularities of distribution , 1954 .

[17]  J. Kiefer.  On large deviations of the empiric D. F. of vector chance variables and a law of the iterated logarithm. , 1961 .

[18]  Wolfgang Ch. Schmid,et al.  Bounds for digital nets and sequences , 1997 .