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Mathematics of Computation

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Pseudo-random numbers. The exact distribution of pairs

Author: U. Dieter
Journal: Math. Comp. 25 (1971), 855-883
MSC: Primary 60E05
MathSciNet review: 0298727
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Abstract: Pseudo-random numbers are usually generated by linear congruential methods. Starting with an integer ${y_0}$, a sequence $\{ {y_i}\}$ is constructed by ${y_{i + 1}} \equiv a{y_i} + r \pmod m, m, a, r$ being integers. The derived fractions ${x_i} \equiv {y_i}/m$ are taken as samples from the uniform distribution on [0, 1). In this paper it is shown that the joint probability distribution of pairs ${x_i},{x_{i + s}}$ can be calculated exactly. Explicit calculations show that this distribution is surprisingly near to the uniform distribution for most ’reasonable’ generators. The best approximation to the uniform distribution on the unit-square is achieved if the continued fraction for ${a^s}$ and m (or ${a^s}$ and m/f) is long.

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Keywords: Multiplicatively generated pseudo-random numbers
Article copyright: © Copyright 1971 American Mathematical Society