On the discrepancy of random subsequences of $\{n\alpha \}$ II

Abstract

Let $\alpha$ be an irrational number, let $X_1,X_2,\dots$ 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 $P(|X_1|>t)\sim c{t}^{-\beta }$, $0<\beta <2$, in Part I of this paper 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 $\beta \gamma <1$, 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.

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• Random walks on the circle and Diophantine approximation