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Lower bounds for the discrepancy of inversive congruential pseudorandom numbers with power of two modulus. (English) Zbl 0762.65001
The discrepancy \(D_{m/2}^{(k)}\) of \(k\)-tuples of consecutive pseudorandom numbers generated by the inversive congruential method with modulus \(m=2^ \omega\) with maximum period length \(m/2\) is studied. It is shown that for a given modulus \(m\) there exist multipliers in the inversive congruential method such that \(D_{m/2}^{(k)}\) is at least of the order of magnitude \(m^{-1/2}\) for all dimensions \(k\geq 2\) and all increments \(b\). Therefore, the upper bound \(D_{m/2}^{(2)}=O(m^{-1/2}(\log m)^ 2)\) is in general best possible up to the logarithmic factor.
Reviewer: V.Burjan (Praha)

MSC:
65C10 Random number generation in numerical analysis
11K45 Pseudo-random numbers; Monte Carlo methods
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References:
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