A Note on the Quality of Random Variates Generated by the Ratio of Uniforms Method

Wolfgang Hörmann


The one-dimensional distribution of pseudo-random numbers generated by the ratio of uniforms methods using linear congruential generators (LCGs) as the source of uniform random numbers is investigated in this paper. Due to the two-dimensional lattice structure of LCGs there is always a comparable large gap without a point in the one-dimensional distribution of any ratio of uniforms method. Lower bounds for these probabilities only depending on the modulus and the Beyer quotient of the LCG are proved for the case that the Cauchy the normal or the exponential distribution are generated. These bounds justify the recommendation not to use the ratio of uniforms method combined with LCGs.

Mathematics Subject Classification: 65C10 (Random Number Generation)

CR Categories and Subject Descriptors: G.3 [Probability and Statistics]: Random number generation

General Terms: Algorithms

Key Words: ratio of uniforms method, linear congruential generator, discrepancy

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© ACM, (1994). This is the author's version of the work. It is posted here by permission of ACM for your personal use. Not for redistribution. The definitive version was published in Trans. Model. Comput. Simul. 4(1), 96-106. http://doi.acm.org/10.1145/174619.174623