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

TL;DR

This paper investigates the precise asymptotic behavior of the discrepancy of random subsequences of the form \\{S_k \\alpha\\} for irrational \\alpha, extending previous results to cases with heavy-tailed distributions and general random walks, including limit distributions.

Contribution

It determines the exact order of discrepancy for \\beta \\gamma<1, finds the limit distribution of the normalized discrepancy, and extends results to general random walks without specific arithmetic conditions.

Findings

Discrepancy order for \\beta \\gamma<1 is established.

Limit distribution of normalized discrepancy is derived.

Results are extended to general random walks with mild convergence assumptions.

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 $P (|X_1|>t)\sim ct^{-\beta}$, $0<\beta<2$, 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.