Moment generating functions and moderate deviation principles for lacunary trigonometric sums

TL;DR

This paper investigates the moment generating functions of lacunary trigonometric sums without structural assumptions, identifying the threshold where arithmetic effects influence behavior, and establishes moderate deviation principles showing Gaussian tail behavior.

Contribution

It provides a detailed analysis of the MGF for lacunary sums under minimal assumptions and determines the precise point where arithmetic structure impacts the sums' behavior.

Findings

Identified the threshold where arithmetic effects influence the MGF.

Proved moderate deviation principles for lacunary trigonometric sums.

Showed Gaussian tail behavior between CLT and LDP regimes.

Abstract

In a recent paper, Aistleitner, Gantert, Kabluchko, Prochno and Ramanan studied large deviation principles (LDPs) for lacunary trigonometric sums $\sum_{n=1}^N \cos(2 \pi n_k x)$, where the sequence $(n_k)_{k \geq 1}$ satisfies the Hadamard gap condition $n_{k+1} / n_k \geq q > 1$ for $k \geq 1$. A crucial ingredient in their work were asymptotic estimates for the moment generating function (MGF) of such sums, which turned out to depend on the fine arithmetic structure of the sequence $(n_k)_{k \geq 1}$ in an intricate way. In the present paper we carry out a detailed study of the MGF for lacunary trigonometric sums (without any structural assumptions on the underlying sequence, other than lacunarity), and we determine the sharp threshold where arithmetic effects start to play a role. As an application, we prove moderate deviation principles for lacunary trigonometric sums, and show that the tail probabilities are in accordance with Gaussian behavior throughout the whole range between the central limit theorem and the LDP regime.