The large deviation behavior of lacunary sums

TL;DR

This paper investigates the large deviation behavior of lacunary sums involving periodic functions and Hadamard gap sequences, establishing large deviation principles with different rate functions depending on the gap sequence's structure.

Contribution

It extends large deviation results to lacunary sums with Hadamard gaps, showing the rate function matches the i.i.d. case for large gaps and differs for geometric progressions.

Findings

Large deviation principle holds for sums with large gaps, rate function matches i.i.d. case.

For geometric progressions, the rate function differs and depends on the function and sequence properties.

The results generalize previous work on lacunary trigonometric sums.

Abstract

We study the large deviation behavior of lacunary sums $(S_n/n)_{n\in \mathbb{N} }$ with $S_n:= \sum_{k=1}^n f(a_kU)$, $n\in\mathbb{N}$, where $U$ is uniformly distributed on $[0,1]$, $(a_k)_{k\in\mathbb{N}}$ is an Hadamard gap sequence, and $f\colon \mathbb{R}\to \mathbb{R} $ is a $1$-periodic, (Lipschitz-)continuous mapping. In the case of large gaps, we show that the normalized partial sums satisfy a large deviation principle at speed $n$ and with a good rate function which is the same as in the case of independent and identically distributed random variables $U_k$, $k\in\mathbb{N}$, having uniform distribution on $[0,1]$. When the lacunary sequence $(a_k)_{k\in\mathbb{N}}$ is a geometric progression, then we also obtain large deviation principles at speed $n$, but with a good rate function that is different from the independent case, its form depending in a subtle way on the interplay between the function $f$ and the arithmetic properties of the gap sequence. Our work generalizes some results recently obtained by Aistleitner, Gantert, Kabluchko, Prochno, and Ramanan [Large deviation principles for lacunary sums, preprint, 2020] who initiated this line of research for the case of lacunary trigonometric sums.