The large deviation behavior of lacunary sums

Abstract

AbstractWe study the large deviation behavior of lacunary sums$$(S_n/n)_{n\in {\mathbb {N}}}$$(Sn/n)n∈Nwith$$S_n:= \sum _{k=1}^nf(a_kU)$$Sn:=∑k=1nf(akU),$$n\in {\mathbb {N}}$$n∈N, whereUis uniformly distributed on [0, 1],$$(a_k)_{k\in {\mathbb {N}}}$$(ak)k∈Nis an Hadamard gap sequence, and$$f:{\mathbb {R}}\rightarrow {\mathbb {R}}$$f:R→Ris 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 speednand with a good rate function which is the same as in the case of independent and identically distributed random variables$$U_k$$Uk,$$k\in {\mathbb {N}}$$k∈N, having uniform distribution on [0, 1]. When the lacunary sequence$$(a_k)_{k\in {\mathbb {N}}}$$(ak)k∈Nis a geometric progression, then we also obtain large deviation principles at speedn, 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 functionfand 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 lacunarytrigonometricsums.