Distribution properties of multiply-with-carry random number generators

We study the multiply-with-carry family of generators proposed by Marsaglia as a generalization of previous add-with-carry families. We define for them an infinite state space and focus our attention on the (finite) subset of recurrent states. This subset will, in turn, split into possibly several subgenerators. We discuss the uniformity of the d-dimensional distribution of the output of these subgenerators over their full period. In order to improve this uniformity for higher dimensions, we propose a method for finding good parameters in terms of the spectral test. Our results are stated in a general context and are applied to a related complementary multiply-with-carry family of generators.