Wasserstein convergence rates for empirical measures of random subsequence of {nα}

Bingyao Wu and Jie-Xiang Zhu

Stochastic Processes and their Applications, 2025, vol. 181, issue C

Abstract: Fix an irrational number α. Let X1,X2,… be independent, identically distributed, integer-valued random variables with characteristic function φ, and let Sn=∑i=1nXi be the partial sums. Consider the random walk {Snα}n≥1 on the torus, where {⋅} denotes the fractional part. We study the long time asymptotic behavior 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 Ambrosio et al. (2019) and the continued fraction representation of the irrational number α.

Keywords: Empirical measure; Diophantine approximation; Random walk; Wasserstein distance; Continued fractions (search for similar items in EconPapers)
Date: 2025
References: Add references at CitEc
Citations:

Downloads: (external link)
http://www.sciencedirect.com/science/article/pii/S0304414924002424
Full text for ScienceDirect subscribers only

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:eee:spapps:v:181:y:2025:i:c:s0304414924002424

Ordering information: This journal article can be ordered from
http://http://www.elsevier.com/wps/find/supportfaq.cws_home/regional
https://shop.elsevie ... _01_ooc_1&version=01

DOI: 10.1016/j.spa.2024.104534

Access Statistics for this article

Stochastic Processes and their Applications is currently edited by T. Mikosch

More articles in Stochastic Processes and their Applications from Elsevier
Bibliographic data for series maintained by Catherine Liu ().