On the Chung-Diaconis-Graham random process
Abstract
Chung, Diaconis, and Graham considered random processes of the form Xn+1=2Xn+bn (mod p) where X0=0, p is odd, and bn for n=0,1,2,... are i.i.d. random variables on -1,0,1. If Pr(bn=-1)= Pr(bn=1)=β and Pr(bn=0)=1-2β, they asked which value of β makes Xn 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=2t-1 to show that for 0<β<=1/2, there is no such value of β.
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