On a new class of multiplicative pseudo-random number generators

Abstract

In the computing literature, there are few detailed analytical studies of the global statistical characteristics of a class of multiplicative pseudo-random number generators.

We comment briefly on normal numbers and study analytically the approximately uniform discrete distribution or (j,ϕ)-normality in the sense of Besicovitch for complete periods of fractional parts {x ₀ λ ₁ ⁱ/p^α} on [0, 1] fori=0, 1,..., (p−1)p^α−1−1, i.e. in current terminology, generators given byx _n+1 ≡λ ₁ x _n mod p^α wheren=0, 1,..., (p−1)p ^α−1−1,p is any odd prime, (x ₀,p)=1,λ ₁ is a primitive root modp ², and α≥1 is any positive integer.

We derive the expectationsE(X, α),E(X ², α),E(X _nX_n+k); the varianceV(X, α), and the serial correlation coefficient ϱk. By means of Dedekind sums and some results of H. Rademacher, we investigate the asymptotic properties of ϱk for various lagsk and integers α≥1 and give numerical illustrations. For the frequently used case α=1, we find comparable results to estimates of Coveyou and Jansson as well as a mathematical demonstration of a so-called “rule of thumb” related to the choice ofλ ₁ for small ϱk.

Due to the number of parameters in this class of generators, it may be possible to obtain increased control over the statistical behavior of these pseudo-random sequences both analytically as well as computationally.