The Serial Test for Linear Congruential Pseudo-random Numbers

Let m>2 and r be integers, let y0 be an integer in the least residue system mod m, and let X be an integer coprime to m with X ^ ± 1 (mod m) and (X ~ l)y0 + r ^ 0 (mod m). A sequence y0,yl9...of integers in the least residue system mod m is generated by the recursion yn+ x = Xyn + r (mod m) for n = 0, 1, . . . . In the homogeneous case r = 0 (mod m), one chooses y0 to be coprime to m. The sequence x0, xx, . . . in the interval [0, 1), defined by xn yjm for n = 0, 1,. . . , is a sequence of linear congruential pseudorandom numbers. The sequence is purely periodic; let r denote its least period. In practice, m is taken to be a large prime or a large power of 2. For a given s > 2, the serial test is set up to determine the amount of statistical dependence among s successive terms in the sequence x0, xl9 .. . . To this end, one considers the s-tuples xn = (xn, xn+1, . . . , xn+s_l), n = 0, 1, . . . , and measures the deviation between the empirical distribution of the first N of these s-tuples and the uniform distribution on [0, l] by the quantity DN introduced in [3], where 1 <N<T. For the homogeneous case, effective estimates for DT were established in [3], [4]. By extending techniques from [2] and [4], we can now handle the general case. Estimates for DN with N < r are of great practical interest because in calculations involving linear congruential pseudo-random numbers one only uses an initial segment of the period and not the full period itself. The number R^\\ m, q) is defined as in [3]. Cg will denote an explicitly known constant depending only on s, whose exact value may be different in each occurrence.