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

Fix an irrational number $\alpha$. Let $X_1,X_2,\dots$ be independent, identically distributed, integer-valued random variables with characteristic function $\phi$, and let $S_n={\sum }_{i=1}^n{X}_i$ be the partial sums. Consider the random walk ${\left\{S_n\alpha \right\}}_{n\ge 1}$ on the torus, where $\left\{\cdot \right\}$ 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 $\alpha$ and the Hölder continuity of the characteristic function $\phi$ 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 $\alpha$.