Lower bounds in spectral tests for vectors of nonsuccessive values produced by multiple recursive generator with some zero multipliers

LʹEcuyer develops lower bounds for the maximum distance of the parallel hyperplanes generated by the vectors of nonsuccessive random numbers (RNs) obtained from multiple recursive generators (MRGs). In this paper, a stronger bound, twice that obtained by LʹEcuyer, is derived. For kth-order MRGs with fewer than k terms in the recursive relationship, much stronger bounds are found. Large distance implies bad lattice structure. In designing RN generators, this factor should be taken into consideration