The large deviation behavior of lacunary sums

Abstract

We study the large deviation behavior of lacunary sums (S n /n) n∈N with S n := n k=1 f (a k U ), n ∈ N, where U is uniformly distributed on [0, 1], (a k ) k∈N is an Hadamard gap sequence, and f : R → 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 ∈ N, having uniform distribution on [0, 1]. When the lacunary sequence (a k ) k∈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.