Stream correlations in multiple recursive and congruential generators

The use of common random numbers is a method of variance reduction usually implemented by allocating different but repeatable random number streams to different distributions in a simulation. Linear or multiplicative congruential generators with linearly related seeds for the streams are commonly used. We show that there is a simple mathematical relationship between the ith number in streams of multiple recursive or congruential generators and that streams produced from linearly related seeds have serious undesirable correlations.