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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

Distribution Properties of
Multiply-with-Carry Random Number Generators


Authors: Raymond Couture and Pierre L’Ecuyer
Journal: Math. Comp. 66 (1997), 591-607
MSC (1991): Primary 65C10
MathSciNet review: 1401939
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Abstract: We study the multiply-with-carry family of generators proposed by Marsaglia as a generalization of previous add-with-carry families. We define for them an infinite state space and focus our attention on the (finite) subset of recurrent states. This subset will, in turn, split into possibly several subgenerators. We discuss the uniformity of the $d$-dimensional distribution of the output of these subgenerators over their full period. In order to improve this uniformity for higher dimensions, we propose a method for finding good parameters in terms of the spectral test. Our results are stated in a general context and are applied to a related complementary multiply-with-carry family of generators.


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Additional Information

Raymond Couture
Affiliation: Département d’Informatique et de Recherche Opérationnelle, Université de Montréal, C.P. 6128, Succ. Centre-Ville, Montréal, H3C 3J7, Canada
Email: couture@iro.umontreal.ca

Pierre L’Ecuyer
Affiliation: Département d’Informatique et de Recherche Opérationnelle, Université de Montréal, C.P. 6128, Succ. Centre-Ville, Montréal, H3C 3J7, Canada
Email: lecuyer@iro.umontreal.ca

DOI: http://dx.doi.org/10.1090/S0025-5718-97-00827-2
PII: S 0025-5718(97)00827-2
Keywords: Random number generation, recurrences with carry, lattice structure
Received by editor(s): November 29, 1995
Received by editor(s) in revised form: April 24, 1996
Additional Notes: This work has been supported by NSERC-Canada grant # OGP0110050 and FCAR-Québec grant # 93ER1654 to the second author. We wish to thank Brian Whitney who brought reference \cite{rMAR94a} to our attention.
Article copyright: © Copyright 1997 American Mathematical Society