2021/07/27 by Joscha Prochno, Frühwirth, Lorenz, Prochno, Joscha +2 · 2 citations
Mathematics · #11D45 #11K70 #11L03 #60F10 #Analytic Number Theory Research #FOS: Mathematics #Mathematical Approximation and Integration #Number Theory (math.NT) #Primary 42A55 #Probability (math.PR) #Secondary 37A05 #advanced mathematical theories
paper · pdf · doi:10.48550/arxiv.2107.12860
openalex publication_date 2021/07/27 · openalex created_date 2022/07/25 · openalex updated_date 2026/07/28
We study the large deviation behavior of lacunary sums (Sn/n)n∈ ℕ with Sn:= ∑k=1n f(akU), n∈ℕ, where U is uniformly distributed on [0,1], (ak)k∈ℕ is an Hadamard gap sequence, and f\colon 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 Uk, k∈ℕ, having uniform distribution on [0,1]. When the lacunary sequence (ak)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.