Wasserstein Convergence Rates for Empirical Measures of Random Subsequence of $\{n\alpha\}$

TL;DR

This paper investigates how quickly the empirical distribution of a random walk on the torus converges to the uniform distribution in Wasserstein distance, highlighting the influence of number-theoretic properties of lpha and the characteristic function.

Contribution

It establishes convergence rates for empirical measures of random walks on the torus, revealing dependence on Diophantine approximation and characteristic function regularity.

Findings

Wasserstein convergence rate depends on lpha's Diophantine properties.

The regularity of the characteristic function affects convergence speed.

A critical phenomenon in convergence behavior is identified.

Abstract

Fix an irrational number $\alpha$. Let $X_1,X_2,\cdots$ be independent, identically distributed, integer-valued random variables with characteristic function $\varphi$, and let $S_n=\sum_{i=1}^n X_i$ be the partial sums. Consider the random walk $\{S_n \alpha\}_{n\ge 1}$ on the torus, where $\{\cdot\}$ denotes the fractional part. We study the long time asymptotic behaviour of the empirical measure of this random walk to the uniform distribution under the general $p$-Wasserstein distance. Our results show that the Wasserstein convergence rate depends on the Diophantine properties of $\alpha$ and the H\"older continuity of the characteristic function $\varphi$ at the origin, and there is an interesting critical phenomenon that will occur. The proof is based on the PDE approach developed by L. Ambrosio, F. Stra and D. Trevisan in [2] and the continued fraction representation of the irrational number $\alpha$.