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A Multiplicatively Symmetrized Version of the Chung-Diaconis-Graham Random Process

Abstract

This paper considers random processes of the form \(X_{n+1}=a_nX_n+b_n\pmod p\) where p is odd, \(X_0=0\), \((a_0,b_0), (a_1,b_1), (a_2,b_2),\ldots \) are i.i.d., and \(a_n\) and \(b_n\) are independent with \(P(a_n=2)=P(a_n=(p+1)/2)=1/2\) and \(P(b_n=1)=P(b_n=0)=P(b_n=-1)=1/3\). This can be viewed as a multiplicatively symmetrized version of a random process of Chung, Diaconis, and Graham. This paper shows that order \((\log p)^2\) steps suffice for \(X_n\) to be close to uniformly distributed on the integers mod p for all odd p while order \((\log p)^2\) steps are necessary for \(X_n\) to be close to uniformly distributed on the integers mod p.

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References

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Acknowledgements

The author would like to thank the referee for some suggestions.

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Correspondence to Martin Hildebrand.

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Hildebrand, M. A Multiplicatively Symmetrized Version of the Chung-Diaconis-Graham Random Process. J Theor Probab 35, 1216–1225 (2022). https://doi.org/10.1007/s10959-021-01088-3

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  • DOI: https://doi.org/10.1007/s10959-021-01088-3

Keywords

  • Chung–Diaconis–Graham random process
  • Random walks
  • Integers mod p

Mathematics Subject Classification (2020)

  • 60B15
  • 60G50