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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.98.

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Pseudo-random numbers. The exact distribution of pairs
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by U. Dieter PDF
Math. Comp. 25 (1971), 855-883 Request permission

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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Additional Information
  • © Copyright 1971 American Mathematical Society
  • Journal: Math. Comp. 25 (1971), 855-883
  • MSC: Primary 60E05
  • DOI: https://doi.org/10.1090/S0025-5718-1971-0298727-8
  • MathSciNet review: 0298727