The large deviation behavior of lacunary sums

Abstract

We study the large deviation behavior of lacunary sums $(S_n/n{)}_{n\in \mathbb{N}}$ with $S_n:={\sum }_{k=1}^{n}f(a_{k}U)$, $n\in \mathbb{N}$, where $U$ is uniformly distributed on $[0,1]$, $(a_k{)}_{k\in \mathbb{N}}$ is an Hadamard gap sequence, and $f:\mathbb{R}\to \mathbb{R}$ 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.