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Average equidistribution and statistical independence properties of digital inversive pseudorandom numbers over parts of the period


Author: Frank Emmerich
Journal: Math. Comp. 71 (2002), 781-791
MSC (2000): Primary 65C10; Secondary 11K45
DOI: https://doi.org/10.1090/S0025-5718-01-01328-X
Published electronically: October 25, 2001
MathSciNet review: 1885628
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Abstract: This article deals with the digital inversive method for generating uniform pseudorandom numbers. Equidistribution and statistical independence properties of the generated pseudorandom number sequences over parts of the period are studied based on the distribution of tuples of successive terms in the sequence. The main result is an upper bound for the average value of the star discrepancy of the corresponding point sets. Additionally, lower bounds for the star discrepancy are established. The method of proof relies on bounds for exponential sums.


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  • 1. J. Eichenauer-Herrmann, Inversive congruential pseudorandom numbers: a tutorial, Internat. Statist. Rev. 60 (1992), 167-176.
  • 2. -, Pseudorandom number generation by nonlinear methods, Internat. Statist. Rev. 63 (1995), 247-255.
  • 3. J. Eichenauer-Herrmann, E. Herrmann, and S. Wegenkittl, A survey of quadratic and inversive congruential pseudorandom numbers, Monte Carlo and Quasi-Monte Carlo Methods 1996 (H. Niederreiter, P. Hellekalek, G. Larcher, and P. Zinterhof, eds.), Lecture Notes in Statistics, vol. 127, Springer, New York, 1998, pp. 66-97. MR 99d:11085
  • 4. J. Eichenauer-Herrmann and H. Niederreiter, Digital inversive pseudorandom numbers, ACM Trans. Modeling and Computer Simulation 4 (1994), 339-349.
  • 5. F. Emmerich, Pseudorandom number and vector generation by compound inversive methods, Thesis, Darmstadt, 1996.
  • 6. -, Statistical independence properties of inversive pseudorandom vectors over parts of the period, ACM Trans. Modeling and Computer Simulation 8 (1998), 140-152.
  • 7. P. Hellekalek, General discrepancy estimates: the Walsh function system, Acta Arith. 67 (1994), 209-218. MR 95h:65003
  • 8. J. Kiefer, On large deviations of the empiric d.f. of vector chance variables and a law of the iterated logarithm, Pacific J. Math. 11 (1961), 649-660. MR 24:A1732
  • 9. R. Lidl and H. Niederreiter, Finite fields, Addison-Wesley, Reading, MA, 1983. MR 86c:11106
  • 10. H. Niederreiter, Random number generation and quasi-Monte Carlo methods, SIAM, Philadelphia, PA, 1992. MR 93h:65008
  • 11. -, Pseudorandom vector generation by the inversive method, ACM Trans. Modeling and Computer Simulation 4 (1994), 191-212.
  • 12. -, New developments in uniform pseudorandom number and vector generation, Monte Carlo and Quasi-Monte Carlo Methods in Scientific Computing (H. Niederreiter and P. J.-S. Shiue, eds.), Lecture Notes in Statistics, vol. 106, Springer, New York, 1995, pp. 87-120. MR 97k:65019
  • 13. -, Improved bounds in the multiple-recursive matrix method for pseudorandom number and vector generation, Finite Fields Appl. 2 (1996), 225-240. MR 97d:11120
  • 14. H. Niederreiter and I. E. Shparlinski, On the distribution of pseudorandom numbers and vectors generated by inversive methods, Appl. Algebra Engrg. Comm. Comput. 10 (2000), 189-202.

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Additional Information

Frank Emmerich
Affiliation: T-Nova Deutsche Telekom Innovationsgesellschaft, Technologiezentrum, Am Kavalleriesand 3, D-64295 Darmstadt, F. R. Germany

DOI: https://doi.org/10.1090/S0025-5718-01-01328-X
Keywords: Uniform pseudorandom numbers, digital inversive method, average equidistribution behaviour, average statistical independence properties, star discrepancy, exponential sums
Received by editor(s): November 10, 1999
Received by editor(s) in revised form: July 12, 2000
Published electronically: October 25, 2001
Article copyright: © Copyright 2001 American Mathematical Society

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