Electronic Communications in Probability

On the Chung-Diaconis-Graham random process

Martin Hildebrand

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Chung, Diaconis, and Graham considered random processes of the form $X_{n+1}=2X_n+b_n \pmod p$ where $X_0=0$, $p$ is odd, and $b_n$ for $n=0,1,2,\dots$ are i.i.d. random variables on $\{-1,0,1\}$. If $\Pr(b_n=-1)=\Pr(b_n=1)=\beta$ and $\Pr(b_n=0)=1-2\beta$, they asked which value of $\beta$ makes $X_n$ 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 $0<\beta\le 1/2$, there is no such value of $\beta$.

Article information

Electron. Commun. Probab., Volume 11 (2006), paper no. 34, 347-356.

Accepted: 15 December 2006
First available in Project Euclid: 4 June 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60B15: Probability measures on groups or semigroups, Fourier transforms, factorization
Secondary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)

Random processes discrete Fourier analysis

This work is licensed under aCreative Commons Attribution 3.0 License.


Hildebrand, Martin. On the Chung-Diaconis-Graham random process. Electron. Commun. Probab. 11 (2006), paper no. 34, 347--356. doi:10.1214/ECP.v11-1237. https://projecteuclid.org/euclid.ecp/1465058877

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