Wasserstein Convergence Rates for Empirical Measures of Random Subsequence of \nα\

2024/11/24 by Bingyao Wu, Wu, Bingyao, Jie-Xiang Zhu +1 · 2 citations
Mathematics · #11J70 #42A05 #60B10 #60G50 #Analytic Number Theory Research #FOS: Mathematics #Point processes and geometric inequalities #Probability (math.PR) #Random Matrices and Applications

paper · pdf · doi:10.48550/arxiv.2411.15724

openalex publication_date 2024/11/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Fix an irrational number α. Let X₁,X₂,⋯ be independent, identically distributed, integer-valued random variables with characteristic function φ, and let S_n=∑_i=1ⁿ X_i be the partial sums. Consider the random walk \S_n α\_{n≥ 1} on the torus, where \⋅\ 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 α and the Hölder continuity of the characteristic function φ 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 α.

Cited by

• Random walks and quadratic number fields
• Exact rate of convergence for the empirical measure of a subordinated process in p-Wasserstein distance

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