Moment generating functions and moderate deviation principles for lacunary trigonometric sums
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866916867675783168 |
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| author | Aistleitner, Christoph Frühwirth, Lorenz Hauke, Manuel Manskova, Maryna |
| author_facet | Aistleitner, Christoph Frühwirth, Lorenz Hauke, Manuel Manskova, Maryna |
| contents | In a recent paper, Aistleitner, Gantert, Kabluchko, Prochno and Ramanan studied large deviation principles (LDPs) for lacunary trigonometric sums $\sum_{n=1}^N \cos(2 πn_k x)$, where the sequence $(n_k)_{k \geq 1}$ satisfies the Hadamard gap condition $n_{k+1} / n_k \geq q > 1$ for $k \geq 1$. 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 $(n_k)_{k \geq 1}$ 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2502_20930 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Moment generating functions and moderate deviation principles for lacunary trigonometric sums Aistleitner, Christoph Frühwirth, Lorenz Hauke, Manuel Manskova, Maryna Probability Classical Analysis and ODEs Number Theory 42A55, 11K06, 42A70, 60F05, 60F10 In a recent paper, Aistleitner, Gantert, Kabluchko, Prochno and Ramanan studied large deviation principles (LDPs) for lacunary trigonometric sums $\sum_{n=1}^N \cos(2 πn_k x)$, where the sequence $(n_k)_{k \geq 1}$ satisfies the Hadamard gap condition $n_{k+1} / n_k \geq q > 1$ for $k \geq 1$. 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 $(n_k)_{k \geq 1}$ 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. |
| title | Moment generating functions and moderate deviation principles for lacunary trigonometric sums |
| topic | Probability Classical Analysis and ODEs Number Theory 42A55, 11K06, 42A70, 60F05, 60F10 |
| url | https://arxiv.org/abs/2502.20930 |