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Random walks on the circle and Diophantine approximation

Istvan Berkes, Bence Borda

Abstract

Random walks on the circle group $\mathbb{R}/\mathbb{Z}$ whose elementary steps are lattice variables with span $α\not\in \mathbb{Q}$ or $p/q \in \mathbb{Q}$ taken mod $\mathbb{Z}$ exhibit delicate behavior. In the rational case we have a random walk on the finite cyclic subgroup $\mathbb{Z}_q$, and the central limit theorem and the law of the iterated logarithm follow from classical results on finite state space Markov chains. In this paper we extend these results to random walks with irrational span $α$, and explicitly describe the transition of these Markov chains from finite to general state space as $p/q \to α$ along the sequence of best rational approximations. We also consider the rate of weak convergence to the stationary distribution in the Kolmogorov metric, and in the rational case observe a surprising transition from polynomial to exponential decay after $\approx q^2$ steps; this seems to be a new phenomenon in the theory of random walks on compact groups. In contrast, the rate of weak convergence to the stationary distribution in the total variation metric is purely exponential.

Random walks on the circle and Diophantine approximation

Abstract

Random walks on the circle group whose elementary steps are lattice variables with span or taken mod exhibit delicate behavior. In the rational case we have a random walk on the finite cyclic subgroup , and the central limit theorem and the law of the iterated logarithm follow from classical results on finite state space Markov chains. In this paper we extend these results to random walks with irrational span , and explicitly describe the transition of these Markov chains from finite to general state space as along the sequence of best rational approximations. We also consider the rate of weak convergence to the stationary distribution in the Kolmogorov metric, and in the rational case observe a surprising transition from polynomial to exponential decay after steps; this seems to be a new phenomenon in the theory of random walks on compact groups. In contrast, the rate of weak convergence to the stationary distribution in the total variation metric is purely exponential.
Paper Structure (11 sections, 13 theorems, 135 equations)

This paper contains 11 sections, 13 theorems, 135 equations.

Table of Contents

  1. Introduction
  2. Related results
  3. Proof of Theorems \ref{['maintheorem']} and \ref{['Wienertheorem']}
  4. A lemma on characteristic functions
  5. An exponential sum
  6. The variance
  7. Approximation by independent variables
  8. Proof of Theorem \ref{['discreteKolmogorovtheorem']}
  9. Upper bounds
  10. Lower bounds
  11. Total variation metric

Key Result

Theorem 1

Assume that $\mathbb E X_1^2 < \infty$, and let $\varphi (x) = \mathbb E e^{ixX_1}$ denote the characteristic function of $X_1$. There exists a constant $q_0$ depending only on the distribution of $X_1$ with the following property. If $p/q$ is a reduced fraction such that $q \ge q_0$, the maximal sp and The implied constants in the lower bounds depend only on the distribution of $X_1$; the implie

Theorems & Definitions (26)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • ...and 16 more