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On the Chung-Diaconis-Graham random process

Martin Hildebrand
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Abstract

Chung, Diaconis, and Graham considered random processes of the form X_{n+1}=2X_n+b_n (mod p) where X_0=0, p is odd, and b_n for n=0,1,2,... are i.i.d. random variables on {-1,0,1}. If Pr(b_n=-1)= Pr(b_n=1)=βand Pr(b_n=0)=1-2β, they asked which value of βmakes X_n get close to uniformly distributed on the integers mod p the slowest. In this paper, we extend the results of Chung, Diaconis, and Graham in the case p=2^t-1 to show that for 0<β<=1/2, there is no such value of β.

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Martin Hildebrand. 2007-08-20. On the Chung-Diaconis-Graham random process. https://arxiv.org/abs/math/0508427

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