Statistical Justification of Combination Generators

The combination generator, first proposed by Wichmann and Hill (1982), is constructed by taking the fractional part of the sum of several random number generators. It is probably one of the most popular random number generators used. Its empirical performance is superior to the classical Lehmer congruential generator. However, its theoretical justification is somewhat primitive. In this paper, we give some theoretical support for such an important generator, from a statistical theory viewpoint. Specifically, we prove that the combination generator method is superior to each component random number generator method, in terms of (1) uniformity and (2) independence.