Congruential generators all are RANDUs!

In case you did not read all the slides of Regis Lebrun’s talk on pseudo-random generators I posted yesterday, one result from Marsaglia’s (in a 1968 PNAS paper) exhibited my ignorance during Regis’ Big’ MC seminar on Thursday. Marsaglia indeed showed that all multiplicative congruential generators

lie on a series of hyperplanes whose number gets ridiculously small as the dimension d increases! If you turn the ‘s into uniforms and look at the d dimensional vectors

they are on a small number of hyperplanes, at most , which gives 41 hyperplanes when … So in this sense all generators share the same poor property as the infamous RANDU which is such that that is always over one of 16 hyperplanes, an exercise we use in both Introducing Monte Carlo Methods with R and Monte Carlo Statistical Methods (but not in our general audience out solution manual). I almost objected to the general result being irrelevant as the ‘s share ‘s, but of course the subsequence also share enjoys this property!