Evaluation of the discrepancy of the linear congruential pseudo-random number sequences (Q1813701)

The authors use the following measure \(D_ n(x_ 1,x_ 2,\dots,x_ n) = \hbox{supremum}\{| N(a,b;x,n)-(b-a)| : 0\leq a<b\leq 1\}\) to appreciate the discrepancy of a given pseudo-random uniform sequence \(x_ 1,x_ 2,\dots,x_ n\) from its real rectangular distribution, where \(N(a,b;x,n)\) is the number of terms \(x_ j\), \(1\leq j\leq n\), which belongs to the interval \([a,b)\). Essentially they compute the maximum of the differences between the empirical frequencies of the sequence \(x_ 1,x_ 2,\dots,x_ n\) into the domains \(D\) and their theoretical probabilities, this for all possible domains \(D\subset [0,1)\). Practically is impossible to consider all domains \(D\). For solving the problem the authors propose three variants which differ by the selection of the domains \(D\). In the first procedure the domains \(D\) are intervals \([a,b]\) where \(a\) and \(b-a\) vary with imposed quantities. The second procedure uses the domains \([0,x_ j]\), \(1\leq j \leq n\). The third procedure considers the domains \([0,k/m]\), \(1\leq k \leq m\), where \(1/m\) is the computation error of the discrepancy. Graphic results are included, too.