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

Lower bounds for the discrepancy of inversive congruential pseudorandom numbers with power of two modulus

Authors: Jürgen Eichenauer-Herrmann and Harald Niederreiter
Journal: Math. Comp. 58 (1992), 775-779
MSC: Primary 65C10; Secondary 11K45
MathSciNet review: 1122066
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Abstract: The inversive congruential method with modulus $ m = {2^\omega }$ for the generation of uniform pseudorandom numbers has recently been introduced. The discrepancy $ D_{m/2}^{(k)}$ of k-tuples of consecutive pseudorandom numbers generated by such a generator with maximal period length $ m/2$ is the crucial quantity for the analysis of the statistical independence properties of these pseudorandom numbers by means of the serial test. It is proved that for a positive proportion of the inversive congruential generators with maximal period length, the discrepancy $ D_{m/2}^{(k)}$ is at least of the order of magnitude $ {m^{ - 1/2}}$ for all $ k \geq 2$. This shows that the bound $ D_{m/2}^{(2)} = O({m^{ - 1/2}}{(\log m)^2})$ established by the second author is essentially best possible.

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PII: S 0025-5718(1992)1122066-X
Keywords: Pseudorandom number generator, inversive congruential method, power of two modulus, discrepancy
Article copyright: © Copyright 1992 American Mathematical Society