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

Abstract

It is well-known that for a quickly increasing sequence $(n_k{)}_{k\ge 1}$ the functions $(\cos 2\pi n_{k}x{)}_{k\ge 1}$ show a behavior which is typical for sequences of independent random variables. If the growth condition on $(n_k{)}_{k\ge 1}$ is relaxed then this almost-independent behavior generally fails. Still, probabilistic constructions show that for \emph{some} very slowly increasing sequences $(n_k{)}_{k\ge 1}$ this almost-independence property is preserved. For example, there exists $(n_k{)}_{k\ge 1}$ 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 $(n_k{)}_{k\ge 1}$ with bounded gaps such that $$\underset{N\to \infty }{\mathrm{limsup}}\frac{\left|{\sum }_{k=1}^N\cos 2\pi n_{k}x\right|}{\sqrt{N\log \log N}}=\infty \text{for almost all x.}$$ 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 $\Lambda \ge 0$ there exists a sequence of integers $(n_k{)}_{k\ge 1}$ 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 $(⟨n_{k}x⟩{)}_{k\ge 1}$.

This manuscript is a complement to the paper "On the law of the iterated logarithm for trigonometric series with bounded gaps", Probab. Th. Rel. Fields, 154 (2012), no. 3-4, 607--620, by the same authors