Wasserstein Convergence Rates for Empirical Measures of Random Subsequence of $\{nα\}$
Fuente:
arXiv
Enregistré dans:
| Auteurs principaux: | , |
|---|---|
| Format: | Preprint |
| Publié: |
2024
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866915032334336000 |
|---|---|
| author | Wu, Bingyao Zhu, Jie-Xiang |
| author_facet | Wu, Bingyao Zhu, Jie-Xiang |
| contents | Fix an irrational number $α$. Let $X_1,X_2,\cdots$ be independent, identically distributed, integer-valued random variables with characteristic function $φ$, and let $S_n=\sum_{i=1}^n X_i$ be the partial sums. Consider the random walk $\{S_n α\}_{n\ge 1}$ 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 $α$ 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 $α$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_15724 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Wasserstein Convergence Rates for Empirical Measures of Random Subsequence of $\{nα\}$ Wu, Bingyao Zhu, Jie-Xiang Probability 60G50, 60B10, 11J70, 42A05 Fix an irrational number $α$. Let $X_1,X_2,\cdots$ be independent, identically distributed, integer-valued random variables with characteristic function $φ$, and let $S_n=\sum_{i=1}^n X_i$ be the partial sums. Consider the random walk $\{S_n α\}_{n\ge 1}$ 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 $α$ 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 $α$. |
| title | Wasserstein Convergence Rates for Empirical Measures of Random Subsequence of $\{nα\}$ |
| topic | Probability 60G50, 60B10, 11J70, 42A05 |
| url | https://arxiv.org/abs/2411.15724 |