A lower bound for the Chung-Diaconis-Graham random process

TL;DR

This paper establishes a lower bound on the number of steps needed for a specific random process to approach uniform distribution, demonstrating that certain processes require more than a logarithmic number of steps.

Contribution

The paper provides a new lower bound for the mixing time of the Chung-Diaconis-Graham process with specific random increments, advancing understanding of its convergence behavior.

Findings

More than c log_2 p steps are needed for the process to become close to uniform.

The process with b_n uniformly distributed among {-1,0,1} does not mix rapidly.

A constant c > 1 exists such that mixing time exceeds c log_2 p.

Abstract

Chung, Diaconis, and Graham considered random processes of the form X_{n+1}=a_n X_n+b_n (mod p) where p is odd, X_0=0, a_n=2 always, and b_n are i.i.d. for n=0,1,2,... . In this paper, we show that if P(b_n=-1)=P{b_n=0)=P(b_n=1)=1/3, then there exists a constant c>1 such that c log_2 p steps are not enough to make X_n get close to uniformly distributed on the integers mod p.