The Annals of Probability

Random Walks Arising in Random Number Generation

F. R. K. Chung, Persi Diaconis, and R. L. Graham

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Random number generators often work by recursively computing $X_{n+1} \equiv aX_n + b (\mod p)$. Various schemes exist for combining these random number generators. In one scheme, $a$ and $b$ are themselves chosen each time from another generator. Assuming that this second source is truly random, we investigate how long it takes for $X_n$ to become random. For example, if $a = 1$ and $b = 0, 1$, or $-1$ each with probability $\frac{1}{3}$, then $cp^2$ steps are required to achieve randomness. On the other hand, if $a = 2$ and $b = 0, 1$, or $-1$, each with probability $\frac{1}{3}$, then $c \log p \log\log p$ steps always suffice to guarantee randomness, and for infinitely many $p$, are necessary as well, although, in fact, for almost all odd $p, 1.02 \log_2 p$ steps are enough.

Article information

Ann. Probab., Volume 15, Number 3 (1987), 1148-1165.

First available in Project Euclid: 19 April 2007

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 60B15: Probability measures on groups or semigroups, Fourier transforms, factorization
Secondary: 60J15

Random walk Fourier analysis discrete Fourier analysis random number generation


Chung, F. R. K.; Diaconis, Persi; Graham, R. L. Random Walks Arising in Random Number Generation. Ann. Probab. 15 (1987), no. 3, 1148--1165. doi:10.1214/aop/1176992088.

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