On the discrepancy of random subsequences of $\{n\alpha \}$, II
Volume 199 / 2021
Abstract
Let $\alpha $ be an irrational number, let $X_1, X_2, \ldots $ be independent, identically distributed, integer-valued random variables, and put $S_k=\sum _{j=1}^k X_j$. Assuming that $X_1$ has finite variance or heavy tails $\mathbb P (|X_1| \gt t)\sim ct^{-\beta }$, $0 \lt \beta \lt 2$, in the first part of this paper [Acta Arith. 191 (2019), 383–415] we proved that up to logarithmic factors, the order of magnitude of the discrepancy $D_N (S_k \alpha )$ of the first $N$ terms of the sequence $\{S_k \alpha \}$ is $O(N^{-\tau })$, where $\tau = \min (1/(\beta \gamma ), 1/2)$ (with $\beta =2$ in the case of finite variances) and $\gamma $ is the strong Diophantine type of $\alpha $. This shows a change of behavior of the discrepancy at $\beta \gamma =2$. In this paper we determine the exact order of magnitude of $D_N (S_k \alpha )$ for $\beta \gamma \lt 1$, and determine the limit distribution of $N^{-1/2} D_N (S_k \alpha )$. We also prove a functional version of these results describing the asymptotic behavior of a wide class of functionals of the sequence $\{S_k \alpha \}$. Finally, we extend our results to the discrepancy of $\{S_k\}$ for general random walks $S_k$ without arithmetic conditions on $X_1$, assuming only a mild polynomial rate on the weak convergence of $\{S_k\}$ to the uniform distribution.
