Random number generators with long period and sound statistical properties

To generate random numbers (RNs) of long period for large scale simulation studies, the usual multiplicative congruential RN generator can be extended to higher order. A multiplicative congruential RN generator of order two with prime modulus 2³¹ – 1 attains a maximal period of (2³¹ – 1)² – 1 when the two multipliers (a1, a2) are chosen properly. By fixing a2 at –742938285, a multiplier recommended for first-order generator in a previous study, approximately 1.1 billion choices of a1 which are able to produce RNs of maximal period are investigated in this paper. Via the spectral test of dimensions up to six, 14 sets of multipliers (a1, a2) exhibit good lattice structure in a global sense with a spectral measure greater than 0.84. Ten of these multipliers also pass a battery of tests for detecting departures from local randomness and homogeneity. Furthermore, the execution time is promising on 32-bit machines. In sum, the second-order generators devised in this paper possess the properties of long period, randomness, homogeneity, repeatability, portability, and efficiency, required for practical use.