Source: Electron. J. Statist. Volume 3
(2009), 1531-1541.
Physical random numbers are not as widely used in Monte Carlo integration as pseudo-random numbers are. They are inconvenient for many reasons. If we want to generate them on the fly, then they may be slow. When we want reproducible results from them, we need a lot of storage. This paper shows that we may construct N=n(n−1)/2 pairwise independent random vectors from n independent ones, by summing them modulo 1 in pairs. As a consequence, the storage and speed problems of physical random numbers can be greatly mitigated. The new vectors lead to Monte Carlo averages with the same mean and variance as if we had used N independent vectors. The asymptotic distribution of the sample mean has a surprising feature: it is always symmetric, but never Gaussian. This follows by writing the sample mean as a degenerate U-statistic whose kernel is a left-circulant matrix. Because of the symmetry, a small number B of replicates can be used to get confidence intervals based on the central limit theorem.
References
Arcones, M. A. and Giné, E. (1993). Limit Theorems for U-Processes., Annals of Probability 21 1494–1542.
Davis, P. J. (1979)., Circulant Matrices. Wiley, New York.
Mathematical Reviews (MathSciNet):
MR543191
Devroye, L. (1986)., Non-uniform Random Variate Generation. Springer.
Mathematical Reviews (MathSciNet):
MR836973
Gregory, G. (1977). Large sample theory for U-statistics and tests of fit., The Annals of Statistics 5 110–123.
Mathematical Reviews (MathSciNet):
MR433669
Hoeffding, W. (1948). A class of statistics with asymptotically normal distribution., Annals of Mathematical Statistics 19 293–325.
Mathematical Reviews (MathSciNet):
MR26294
Hong, H. S., Hickernell, F. J. and Wei, G. (2003). The distributio nof the discrepancy of scrambled digital, (t,m,s)–nets. Mathematics and Computers in Simulation 62 335–345.
Karner, H., Schneid, J. and Ueberhuber, C. W. (2003). Spectral decomposition of real circulant matrices., Linear Algebra and its Applications 367 301–311.
L’Ecuyer, P. (2009). Pseudorandom number generators. In, Encyclopedia of quantitative finance ( R. Cont, ed.) Wiley, New York.
Loh, W.-L. (2003). On the asymptotic distribution of scrambled net quadrature., Annals of Statistics 31 1282–1324.
Matoušek, J. (1998). On the, L2–discrepancy for anchored boxes. Journal of Complexity 14 527–556.
Owen, A. B. (1995). Randomly Permuted, (t,m,s)-Nets and (t,s)-Sequences. In Monte Carlo and Quasi-Monte Carlo Methods in Scientific Computing ( H. Niederreiter and P. J.-S. Shiue, eds.) 299–317. Springer-Verlag, New York.
Romano, J. P. and Siegel, A. F. (1986)., Counterexamples in probability and statistics. Wadsworth and Brooks/Cole, Belmont CA.
Mathematical Reviews (MathSciNet):
MR831223
Serfling, R. J. (1980)., Approximation theorems of mathematical statistics. Wiley, New York.
Mathematical Reviews (MathSciNet):
MR595165
van der Mee, C., Rodriguez, G. and Seatzu, S. (2006). Fast superoptimal preconditioning of multiindex Toeplitz matrices., Linear Algebra and its Applications 418 576–590.
