Owen, Art B. Recycling physical random numbers. (English) Zbl 1326.65014 Electron. J. Stat. 3, 1531-1541 (2009). Summary: 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. Cited in 1 Document MSC: 65C05 Monte Carlo methods 60F05 Central limit and other weak theorems 62E20 Asymptotic distribution theory in statistics × Cite Format Result Cite Review PDF Full Text: DOI Euclid References: [1] Arcones, M. A. and Giné, E. (1993). Limit Theorems for U-Processes., Annals of Probability 21 1494-1542. · Zbl 0789.60031 · doi:10.1214/aop/1176989128 [2] Davis, P. J. (1979)., Circulant Matrices . Wiley, New York. · Zbl 0418.15017 [3] Devroye, L. (1986)., Non-uniform Random Variate Generation . Springer. · Zbl 0593.65005 [4] Gregory, G. (1977). Large sample theory for U-statistics and tests of fit., The Annals of Statistics 5 110-123. · Zbl 0371.62033 · doi:10.1214/aos/1176343744 [5] Hoeffding, W. (1948). A class of statistics with asymptotically normal distribution., Annals of Mathematical Statistics 19 293-325. · Zbl 0032.04101 · doi:10.1214/aoms/1177730196 [6] 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. · Zbl 1020.65009 · doi:10.1016/S0378-4754(02)00238-0 [7] Karner, H., Schneid, J. and Ueberhuber, C. W. (2003). Spectral decomposition of real circulant matrices., Linear Algebra and its Applications 367 301-311. · Zbl 1021.15007 · doi:10.1016/S0024-3795(02)00664-X [8] L’Ecuyer, P. (2009). Pseudorandom number generators. In, Encyclopedia of quantitative finance ( R. Cont, ed.) Wiley, New York. [9] Loh, W.-L. (2003). On the asymptotic distribution of scrambled net quadrature., Annals of Statistics 31 1282-1324. · Zbl 1105.62304 · doi:10.1214/aos/1059655914 [10] Matoušek, J. (1998). On the, L 2 -discrepancy for anchored boxes. Journal of Complexity 14 527-556. · Zbl 0942.65021 · doi:10.1006/jcom.1998.0489 [11] 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. · Zbl 0831.65024 [12] Romano, J. P. and Siegel, A. F. (1986)., Counterexamples in probability and statistics . Wadsworth and Brooks/Cole, Belmont CA. · Zbl 0587.60001 [13] Serfling, R. J. (1980)., Approximation theorems of mathematical statistics . Wiley, New York. · Zbl 0538.62002 [14] 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. · Zbl 1111.65044 · doi:10.1016/j.laa.2006.02.034 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.