Moment generating functions and moderate deviation principles for lacunary sums

• 成果类型: Article; Early Access
• 署名作者: Aistleitner, Christoph; Fruhwirth, Lorenz; Hauke, Manuel; Manskova, Maryna
• 署名单位: Graz University of Technology; University of Passau; Norwegian University of Science & Technology (NTNU)
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-025-01405-5
• 发表日期: 2025
• 关键词: limit-theorems

摘要

In a recent paper, Aistleitner, Gantert, Kabluchko, Prochno and Ramanan studied large deviation principles (LDPs) for lacunary trigonometric sums Sigma(N)(K=1 )cos(2 pi n(k)x), where the sequence (n(k)) k >= 1 satisfies the Hadamard gap condition n(k+1)/nk >= q >1 for k >= 1. A crucial ingredient in their work were asymptotic estimates for the moment generating function (MGF) of such sums, which turned out to depend on the fine arithmetic structure of the sequence (n(k)) k >= 1 in an intricate way. In the present paper we carry out a detailed study of the MGF for lacunary trigonometric sums (without any structural assumptions on the underlying sequence, other than lacunarity), and we determine the sharp threshold where arithmetic effects start to play a role. As an application, we prove moderate deviation principles for lacunary trigonometric sums, and show that the tail probabilities are in accordance with Gaussian behavior throughout the whole range between the central limit theorem and the LDP regime.