Average equidistribution and statistical independence properties of digital inversive pseudorandom numbers over parts of the period (Q2781225)

The author of this interesting paper concentrates on digital inversive pseudorandom numbers generators introduced by \textit{J. Eichenauer-Herrmann} and \textit{H. Niederreiter} [ACM Trans. Model. Comput. Simul. 4, 339-349 (1994; Zbl 0847.11038)]. After summarizing some known results and auxiliary results, he shows in Theorem~1 that in the digital inversive method the star discrepancy, on the average over the parameter \(\kappa\), has an order of magnitude at most \(N^{-1/2}(\log q)^s\) for any parameters \(\alpha\) and \(\beta\) defining the generator, provided the condition for the maximum possible period length is met. Moreover, it is shown that for any fixed parameters \(\alpha\) and \(\beta\), only a small percentage of the values of the parameter \(\kappa\) may lead to a star discrepancy with an order of magnitude greater than \(N^{-1/2}(\log q)^s\). On the other hand, Theorem~2 shows that, for any \(\alpha\) and \(\beta\), there exists values of the parameter \(\kappa\) such that the star discrepancy is of an order of magnitude at least \(N^{1/2}\), if \(N\) is not too close to \(q\).