The Marsaglia Random Number Generator

The Marsaglia random number generator [Marsaglia et al. 1990] is known as a combination generator because it is constructed from two different generators. It has a period of about 

The first generator is a lagged Fibonacci generator that performs the following binary 

The values x and y are chosen from numbers earlier in the sequence, so that the th value in the sequence is calculated by  

The second generator is an arithmetic sequence method that generates random numbers using the following mathematical operation  

The c sequence requires one initial seed value and the x sequence requires 97 initial seeds (which should themselves be reasonably random). These can be supplied by the user but in the published algorithm these 97 values were obtained from another combination generator comprising a lagged Fibonacci generator and a congruential algorithm. 

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G. Marsaglia, A Current View of Random Number Generators, in Computing Science and Statistics Proceedings of the XVIth Symposium on the Interface, 1985, pp. 3-10. 

However, a single statistical test is not adequate to verify the randomness of a sequence, because typical MCMC applications can be sensitive to various types of correlations. However, if the generator passes a wide variety of tests, then our confidence in its randomness increases. The tests suggested by Knuth [7] and those implemented in the DIEHARD package by Marsaglia [36] are a standard. Since there are many generators that pass these tests, there is no reason to consider one that is known to fail. (Of course, any generator will eventually fail most tests, so we always must state how many numbers were used in the test level of accuracy). 

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