Weak limits for the diaphony
In one dimension, we represent the diaphony as the L2-norm of a random process which is found to converge weakly to a second order stationary Gaussian; up to scaling, this implies the asymptotic distributions of the diaphony and the *-discrepancy to coincide. Further, we show that properly normalized, the diaphony of n points in dimension d is asymptotically Gaussian if both n and d increase with a certain rate.
KeywordsCentral Limit Theorem Gaussian Process Weak Convergence Asymptotic Distribution Explicit Representation
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- [He196]P. Hellekalek. On correlation analysis of pseudorandom numbers. submitted to the Proceedings of the Second International Conference on Monte Carlo and Quasi-Monte Carlo Methods in Scientific Computing, 1996.Google Scholar
- P. Hall and C.C. Heyde. Martingale Limit Theory and its Application. Probability and Mathematical Statistics. Academic Press, Inc., San Diego, California, 1980.Google Scholar
- [HL96]P. Hellekalek and H. Leeb. Dyadic diaphony. Acta Arith.,to appear, 1996.Google Scholar
- [JHK96]F. James, J. Hoogland, and R. Kleiss. Multidimensional sampling for simulation and integration: measures, discrepancies, and quasi-random numbers. to appear in Comp. Phys. Comm., 1996.Google Scholar
- [Kle96]R. Kleiss. private communications, 1996.Google Scholar
- [Lee96]H. Leeb. The asymptotic distribution of diaphony in one dimension. G-96–52, GERAD - École des Hautes Etudes Commerciales, Montréal, 1996.Google Scholar