Gaussian limits for discrepancies I. Asymptotic results

Abstract We consider the problem of finding, for a given quadratic measure of non-uniformity of a set of N points (such as L2 star-discrepancy or diaphony), the asymptotic distribution of this discrepancy for truly random points in the limit N → ∞. We then examine the circumstances under which this distribution approaches a normal distribution. For large classes of non-uniformity measures, a ‘Central Limit Theorem’ can be derived.

[1]  Hannes Leeb,et al.  Weak limits for the diaphony , 1998 .

[2]  Jiri Hoogland,et al.  Discrepancy - based error estimates for quasi - Monte Carlo , 1996 .

[3]  F. James,et al.  Multidimensional sampling for simulation and integration: measures, discrepancies, and quasi-random numbers , 1996, hep-ph/9606309.

[4]  Jiri Hoogland,et al.  Discrepancy-based error estimates for Quasi-Monte Carlo. I. General formalism , 1996 .

[5]  Robert F. Tichy,et al.  Sequences, Discrepancies and Applications , 1997 .

[6]  R. Kleiss,et al.  Discrepancy-based error estimates for Quasi-Monte Carlo. 2: Applications in one dimension , 1996, hep-ph/9603211.

[7]  Lauwerens Kuipers,et al.  Uniform distribution of sequences , 1974 .

[8]  H. Wozniakowski.  Average case complexity of multivariate integration , 1991 .

[9]  Michel Loève,et al.  Probability Theory I , 1977 .

[10]  E. Wright,et al.  An Introduction to the Theory of Numbers , 1939 .

[11]  S. Tezuka,et al.  Toward real-time pricing of complex financial derivatives , 1996 .

[12]  R. Kleiss.  Average-case complexity distributions: a generalization of the Woz̀niakowski lemma for multidimensional numerical integration , 1992 .

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

[14]  Shu Tezuka,et al.  Polynomial arithmetic analogue of Halton sequences , 1993, TOMC.

[15]  R. Kleiss,et al.  Gaussian limits for discrepancies , 1998 .

[16]  C. Schlier,et al.  Monte Carlo integration with quasi-random numbers: experience with discontinuous integrands , 1997 .

[17]  Spassimir H. Paskov,et al.  Average Case Complexity of Multivariate Integration for Smooth Functions , 1993, J. Complex..