Moderate deviation principles and Mod-Gaussian convergence for lacunary trigonometric sums

TL;DR

This paper investigates moderate deviation principles and mod-Gaussian convergence for lacunary trigonometric sums, revealing that their behavior on certain scales resembles sums of independent variables despite dependencies.

Contribution

It introduces correlation graphs and cumulant methods to establish moderate deviation principles and analyzes mod-Gaussian convergence for lacunary sums, filling a gap between CLT and large deviations.

Findings

No arithmetic effects between CLT and near large deviation scales.

Moderate deviations follow a universal pattern unaffected by arithmetic structure.

Established mod-Gaussian convergence for lacunary trigonometric sums.

Abstract

Classical works of Kac, Salem and Zygmund, and Erd\H{o}s and G\'{a}l have shown that lacunary trigonometric sums despite their dependency structure behave in various ways like sums of independent and identically distributed random variables. For instance, they satisfy a central limit theorem and a law of the iterated logarithm. Those results have only recently been complemented by large deviation principles by Aistleitner, Gantert, Kabluchko, Prochno, and Ramanan, showing that interesting phenomena occur on the large deviation scale that are not visible in the classical works. This raises the question on what scale such phenomena kick in. In this paper, we provide a first step towards a resolution of this question by studying moderate deviation principles for lacunary trigonometric sums. We show that no arithmetic affects are visible between the CLT scaling $\sqrt{n}$ and a scaling $n/\log(n)$ that is only a logarithmic gap away from the large deviations scale. To obtain our results, inspired by the notion of a dependency graph, we introduce correlation graphs and use the method of cumulants. In this work we also obtain results on the mod-Gaussian convergence using different tools.