Deterministic simulation of random processes
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References
- Hermann Weyl, Über die Gleichverteilung von Zahlen mod. Eins, Math. Ann. 77 (1916), no. 3, 313–352 (German). MR 1511862, DOI 10.1007/BF01475864
- J. G. van der Corput, Diophantische Ungleichungen. I. Zur Gleichverteilung Modulo Eins, Acta Math. 56 (1931), no. 1, 373–456 (German). MR 1555330, DOI 10.1007/BF02545780
- J. F. Koksma, Ein mengentheoretischer Satz über die Gleichverteilung modulo Eins, Compositio Math. 2 (1935), 250–258 (German). MR 1556918 J. F. Koksma, Diophantische Approximationen, Chelsea, New York, 1936. D. L. Jagerman, The Autocorrelation Function of a Sequence Uniformly Distributed Modulo One, System Development Corp. Report SP-148, 1960, to appear in Annals of Mathematical Statistics.
- Olga Taussky and John Todd, Generation and testing of pseudo-random numbers, Symposium on Monte Carlo methods, University of Florida, 1954, John Wiley and Sons, Inc., New York; Chapman and Hall, Ltd., London, 1956, pp. 15–28. MR 0080382
- J. N. Franklin, On the equidistribution of pseudo-random numbers, Quart. Appl. Math. 16 (1958), 183–188. MR 93501, DOI 10.1090/S0033-569X-1958-93501-2
- A. Zygmund, Trigonometric series. 2nd ed. Vols. I, II, Cambridge University Press, New York, 1959. MR 0107776 G. Birkhoff & S. MacLane, A Survey of Modern Algebra, Macmillan, 1948.
- C. B. Bell, On the structure of distribution-free statistics, Ann. Math. Statist. 31 (1960), 703–709. MR 119310, DOI 10.1214/aoms/1177705796
- W. F. Bauer, The Monte Carlo method, J. Soc. Indust. Appl. Math. 6 (1958), 438–451. MR 98455 G. E. P. Box & M. E. Mueller, “A note on the generation of random normal deviates,” Ann. Math. Statist., v. 29, 1958, p. 610-611. G. W. Brown, “History of Rand’s random digits—summary,” Monte Carlo Method, Nat. Bur. Standards.; App. Math. Ser. 12, June 11, 1951.
- Eve Bofinger, On a periodic property of pseudo-random sequences, J. Assoc. Comput. Mach. 5 (1958), 261–265. MR 131949, DOI 10.1145/320932.320937
- E. D. Cashwell and C. J. Everett, A practical manual on the Monte Carlo method for random walk problems. , International Tracts in Computer Science and Technology and Their Application, Vol. 1, Pergamon Press, New York-London-Paris-Los Angeles, 1959. MR 0106538
- J. Certaine, On sequences of pseudo-random numbers of maximal length, J. Assoc. Comput. Mach. 5 (1958), 353–356. MR 127511, DOI 10.1145/320941.320949
- Charles E. Clark, The utility of statistics of random numbers, Operations Res. 8 (1960), 185–195. MR 115262, DOI 10.1287/opre.8.2.185
- R. R. Coveyou, Serial correlation in the generation of pseudo-random numbers, J. Assoc. Comput. Mach. 7 (1960), 72–74. MR 117869, DOI 10.1145/321008.321018
- A. P. Dempster, Random allocation designs. I. On general classes of estimation methods, Ann. Math. Statist. 31 (1960), 885–905. MR 125712, DOI 10.1214/aoms/1177705665
- H. J. A. Duparc, C. G. Lekkerkerker, and W. Peremans, Reduced sequences of integers and pseudo-random numbers, Math. Centrum, Amsterdam, 1953. Rapport ZW 1953-002,. MR 0053417
- A. R. Edmonds, The generation of pseudo-random numbers on electronic digital computers, Comput. J. 2 (1959/60), 181–185. MR 112271, DOI 10.1093/comjnl/2.4.181
- D. I. Golenko, Calculation of the characteristics of certain stochastic processes by the Monte Carlo method, Vyčisl. Mat. 5 (1959), 93–108. MR 0114287
- I. J. Good, The serial test for sampling numbers and other tests for randomness, Proc. Cambridge Philos. Soc. 49 (1953), 276–284. MR 60786, DOI 10.1017/s030500410002836x
- Bert F. Green Jr., J. E. Keith Smith, and Laura Klem, Empirical tests of an additive random number generator, J. Assoc. Comput. Mach. 6 (1959), 527–537. MR 107957, DOI 10.1145/320998.321006
- Martin Greenberger, An a priori determination of serial correlation in computer generated random numbers, Math. Comp. 15 (1961), 383–389. MR 144489, DOI 10.1090/S0025-5718-1961-0144489-8 M. Greenberger, “Notes on a new pseudo-random number generator,” J. Assoc. Comput. Mach., v. 8, no. 2, 1961, p. 163-167. M. Greenberger, “Random number generators,” preprints of the Fourteenth National Conference of the ACM, September 1959. P. C. Hammer, “The mid-square method of generating digits,” Monte Carlo Method, National Bureau of Standards, Washington, D. C, 1951, p. 33.
- D. G. N. Hunter, Note on a test for repeating cycles in a pseudo-random number generator, Comput. J. 3 (1960/61), 9. MR 111118, DOI 10.1093/comjnl/3.1.9
- Masatugu Isida and Hiroji Ikeda, Random number generator, Ann. Inst. Statist. Math., Tokyo 8 (1956), 119–126. MR 0086369, DOI 10.1007/bf02863577
- D. L. Johnson, Generating and testing pseudo random numbers on the IBM Type 701, Math. Tables Aids Comput. 10 (1956), 8–13. MR 76467, DOI 10.1090/S0025-5718-1956-0076467-X
- M. L. Juncosa, Random number generation on the BRL high-speed computing machines, Ballistic Research Laboratories, Aberdeen Proving Ground, Md., 1953. Rep. No. 855,. MR 0059617 H. Kahn, Applications of Monte Carlo, Office of Technical Services, Dept. of Commerce, Washington 25, D. C., revised 1956.
- Maurice G. Kendall, The Advanced Theory of Statistics. Volume II, Charles Griffin & Co., Ltd., 42 Drury Lane, London, W.C.2, 1946. MR 0019869 J. E. Kokie, “One statement Fortran-type random number generators,” Paper No. 59 contributed to Fifteenth ACM Nat. Conf., Milwaukee, Wis., August 1960.
- Heidi G. Kuehn, A $48$-bit pseudo-random number generator, Comm. ACM 4 (1961), 350–352. MR 0144490, DOI 10.1145/366678.366690
- D. H. Lehmer, Mathematical methods in large-scale computing units, Proceedings of a Second Symposium on Large-Scale Digital Calculating Machinery, 1949, Harvard University Press, Cambridge, Mass., 1951, pp. 141–146. MR 0044899
- G. Marsaglia, Expressing a random variable in terms of uniform random variables, Ann. Math. Statist. 32 (1961), 894–898. MR 130090, DOI 10.1214/aoms/1177704983
- G. Marsaglia, Generating exponential random variables, Ann. Math. Statist. 32 (1961), 899–900. MR 130091, DOI 10.1214/aoms/1177704984
- Jack Moshman, The generation of pseudo-random numbers on a decimal calculator, J. Assoc. Comput. Mach. 1 (1954), 88–91. MR 61882, DOI 10.1145/320772.320775 M. E. Muller, Generation of Normal Deviates, TR 13, Statistical Techniques Res. Group, of Mathematics, Princeton University.
- Mervin E. Muller, A comparison of methods for generating normal deviates on digital computers, J. Assoc. Comput. Mach. 6 (1959), 376–383. MR 105785, DOI 10.1145/320986.320992
- Mervin E. Muller, An inverse method for the generation of random normal deviates on large-scale computers, Math. Tables Aids Comput. 12 (1958), 167–174. MR 102905, DOI 10.1090/S0025-5718-1958-0102905-1
- E. S. Page, Pseudo-random elements for computers, Appl. Statist. 8 (1959), 124–131. MR 107956, DOI 10.2307/2985548
- A. Rotenberg, A new pseudo-random number generator, J. Assoc. Comput. Mach. 7 (1960), 75–77. MR 117868, DOI 10.1145/321008.321019
- I. M. Sobol′, Pseudo-random numbers for the machine “Strela”, Teor. Veroyatnost. i Primenen. 3 (1958), 205–211 (Russian, with English summary). MR 0094923 W. W. Stone, “Team: A three-dimensional Monte Carlo reactor program utilizing a fast random variable generation technique,” paper No. 60 contributed to Fifteenth ACM Nat. Conf., Milwaukee, Wis., Aug. 1960. J. Todd, “Some experiments on Monte Carlo method,” presented before the Inst. of Math. Statistics, Boston, 1951. J. von Neuman, “Various techniques in connection with random digits,” Monte Carlo Methods, Nat. Bur. of Standards, Appl. Math. Ser. v. 12, 1951, p. 36-38.
- Neal Zierler, Linear recurring sequences, J. Soc. Indust. Appl. Math. 7 (1959), 31–48. MR 101979
Additional Information
- © Copyright 1963 American Mathematical Society
- Journal: Math. Comp. 17 (1963), 28-59
- MSC: Primary 65.15
- DOI: https://doi.org/10.1090/S0025-5718-1963-0149640-3
- MathSciNet review: 0149640