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July, 1987 Random Walks Arising in Random Number Generation
F. R. K. Chung, Persi Diaconis, R. L. Graham
Ann. Probab. 15(3): 1148-1165 (July, 1987). DOI: 10.1214/aop/1176992088


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.


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F. R. K. Chung. Persi Diaconis. R. L. Graham. "Random Walks Arising in Random Number Generation." Ann. Probab. 15 (3) 1148 - 1165, July, 1987.


Published: July, 1987
First available in Project Euclid: 19 April 2007

zbMATH: 0622.60016
MathSciNet: MR893921
Digital Object Identifier: 10.1214/aop/1176992088

Primary: 60B15
Secondary: 60J15

Keywords: discrete Fourier analysis , Fourier analysis , random number generation , Random walk

Rights: Copyright © 1987 Institute of Mathematical Statistics


Vol.15 • No. 3 • July, 1987
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