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Random Processes of the Form Xn+1 = anX n + bn (mod p)

Martin Hildebrand
The Annals of Probability
Vol. 21, No. 2 (Apr., 1993), pp. 710-720
Stable URL: http://www.jstor.org/stable/2244672
Page Count: 11
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Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.
Random Processes of the Form Xn+1 = anX
n + bn (mod p)
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Abstract

This paper considers random processes of the form Xn + 1 = anX n + bn (mod p), where X0 = 0 and the sequences an and bn are independent with an identically distributed for n = 0, 1, 2, ... and bn identically distributed for n = 0, 1, 2, .... Chung, Diaconis and Graham studied such processes where an = 2 always; this paper considers more general distributions for an and bn. The question is how long does it take these processes to get close to the uniform distribution? If an is a distribution on Z+ which does not vary with p and bn is a distribution on Z which also does not vary with p, an upper bound on this time is O((log p)2) with appropriate restrictions on p unless an = 1 always, bn = 0 always or an and bn can each take on only one value. This paper uses a recursive relation involving the discrete Fourier transform to find the bound. Under more restrictive conditions for an and bn, this paper finds that a generalization of the technique of Chung, Diaconis and Graham shows that O(log p log log p) steps suffice.

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