On the law of the iterated logarithm for trigonometric series with bounded gaps II

TL;DR

This paper demonstrates that for sequences with bounded gaps, the law of the iterated logarithm (LIL) can exhibit any prescribed limsup behavior, extending understanding of almost-independence in trigonometric series.

Contribution

It proves that any limsup value in the LIL can be achieved for sequences with gaps of size 1 or 2, generalizing previous results and exploring the limits of probabilistic behavior in such sequences.

Findings

Any non-negative real limsup value is attainable in the LIL for sequences with gaps of 1 or 2.

Constructed sequences with bounded gaps can replicate any prescribed LIL limsup behavior.

Results extend to sums of functions and discrepancy of sequences.

Abstract

It is well-known that for a quickly increasing sequence $(n_k)_{k \geq 1}$ the functions $(\cos 2 \pi n_k x)_{k \geq 1}$ show a behavior which is typical for sequences of independent random variables. If the growth condition on $(n_k)_{k \geq 1}$ is relaxed then this almost-independent behavior generally fails. Still, probabilistic constructions show that for \emph{some} very slowly increasing sequences $(n_k)_{k \geq 1}$ this almost-independence property is preserved. For example, there exists $(n_k)_{k \geq 1}$ having bounded gaps such that the normalized sums $\sum \cos 2 \pi n_k x$ satisfy the central limit theorem (CLT). However, due to a ``loss of mass'' phenomenon the variance in the CLT for a sequence with bounded gaps is always smaller than $1/2$. In the case of the law of the iterated logarithm (LIL) the situation is different; as we proved in an earlier paper, there exists $(n_k)_{k \geq 1}$ with bounded gaps such that $$ \limsup_{N \to \infty} \frac{\left| \sum_{k=1}^N \cos 2 \pi n_k x \right|}{\sqrt{N \log \log N}} = \infty \qquad \textrm{for almost all $x$.} $$ In the present paper we prove a complementary results showing that any prescribed limsup-behavior in the LIL is possible for sequences with bounded gaps. More precisely, we show that for any real number $\Lambda \geq 0$ there exists a sequence of integers $(n_k)_{k \geq 1}$ satisfying $n_{k+1} - n_{k} \in \{1,2\}$ such that the limsup in the LIL equals $\Lambda$ for almost all $x$. Similar results are proved for sums $\sum f(n_k x)$ and for the discrepancy of $(\langle n_k x \rangle)_{k \geq 1}$.