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Inversive congruential pseudorandom numbers: distribution of triples

Authors: Jürgen Eichenauer-Herrmann and Harald Niederreiter
Journal: Math. Comp. 66 (1997), 1629-1644
MSC (1991): Primary 65C10; Secondary 11K45
MathSciNet review: 1423072
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Abstract: This paper deals with the inversive congruential method with power of two modulus $m$ for generating uniform pseudorandom numbers. Statistical independence properties of the generated sequences are studied based on the distribution of triples of successive pseudorandom numbers. It is shown that, on the average over the parameters in the inversive congruential method, the discrepancy of the corresponding point sets in the unit cube is of an order of magnitude between $m^{-1/2}$ and $m^{-1/2}(\log m)^3$. The method of proof relies on a detailed discussion of the properties of certain exponential sums.

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Jürgen Eichenauer-Herrmann
Affiliation: Fachbereich Mathematik, Technische Hochschule, Schloßgartenstraße 7, D–64289 Darmstadt, Germany

Harald Niederreiter
Affiliation: Institut für Informationsverarbeitung, Österr. Akademie der Wissenschaften, Sonnenfelsgasse 19, A–1010 Wien, Austria

Keywords: Uniform pseudorandom numbers, inversive congruential method, statistical independence, discrepancy of triples, exponential sums
Received by editor(s): April 12, 1996
Received by editor(s) in revised form: August 23, 1996
Article copyright: © Copyright 1997 American Mathematical Society