Upper bounds in spectral test for multiple recursive random number generators with missing terms

One method for generating random numbers (RNs) of long period recommended by many scholars is the multiple recursive generator (MRG), in that the current RN is essentially a linear combination of the k preceding ones. In this paper, the upper bounds for a figure of merit adopted in the spectral test are derived for the kth order MRG with p ≤ k terms. As p gets smaller, the bounds become smaller as well. The simplest form of the kth order MRG with two terms frequently discussed in literature is found to have the worst bound.