L'Ecuyer 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 L'Ecuyer, 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.
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