Electronic Communications in Probability

On the Chung-Diaconis-Graham random process

Martin Hildebrand

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Abstract

Chung, Diaconis, and Graham considered random processes of the form $X_{n+1}=2X_n+b_n \pmod p$ 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

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

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

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1465058877

Digital Object Identifier
doi:10.1214/ECP.v11-1237

Mathematical Reviews number (MathSciNet)
MR2274529

Zentralblatt MATH identifier
1132.60006

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

Keywords
Random processes discrete Fourier analysis

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

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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References

  • Chung, F. R. K.; Diaconis, Persi; Graham, R. L. Random walks arising in random number generation. Ann. Probab. 15 (1987), no. 3, 1148–1165.
  • Diaconis, Persi. Group representations in probability and statistics. 11. Institute of Mathematical Statistics, Hayward, CA, 1988. vi+198 pp. ISBN: 0-940600-14-5
  • Hildebrand, Martin. Random processes of the form $X_{n+1}=a_n X_n +b_n\pmod p$ Ann. Probab. 21 (1993), no. 2, 710–720.
  • Hildebrand, Martin. Random processes of the form $X_{n+1}=a_n X_n +b_n\pmod p$ 153–174, IMA Vol. Math. Appl., 76, Springer, New York, 1996.