Large deviation principles for lacunary sums
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- by Christoph Aistleitner, Nina Gantert, Zakhar Kabluchko, Joscha Prochno and Kavita Ramanan;
- Trans. Amer. Math. Soc. 376 (2023), 507-553
- DOI: https://doi.org/10.1090/tran/8788
- Published electronically: October 14, 2022
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Abstract:
Let $(a_k)_{k\in \mathbb N}$ be an increasing sequence of positive integers satisfying the Hadamard gap condition $a_{k+1}/a_k> q >1$ for all $k\in \mathbb N$, and let \begin{equation*} S_n(\omega ) = \sum _{k=1}^n \cos (2\pi a_k \omega ), \qquad n\in \mathbb N, \; \omega \in [0,1]. \end{equation*} Then $S_n$ is called a lacunary trigonometric sum, and can be viewed as a random variable defined on the probability space $\Omega = [0,1]$ endowed with Lebesgue measure. Lacunary sums are known to exhibit several properties that are typical for sums of independent random variables. For example, a central limit theorem for $(S_n)_{n\in \mathbb {N}}$ has been obtained by Salem and Zygmund, while a law of the iterated logarithm is due to Erdős and Gál. In this paper we study large deviation principles for lacunary sums. Specifically, under the large gap condition $a_{k+1}/a_k \to \infty$, we prove that the sequence $(S_n/n)_{n \in \mathbb {N}}$ does indeed satisfy a large deviation principle with speed $n$ and the same rate function $\widetilde {I}$ as for sums of independent random variables with the arcsine distribution. On the other hand, we show that the large deviation principle may fail to hold when we only assume the Hadamard gap condition. However, we show that in the special case when $a_k= q^k$ for some $q\in \{2,3,\ldots \}$, $(S_n/n)_{n \in \mathbb {N}}$ satisfies a large deviation principle (with speed $n$) and a rate function $I_q$ that is different from $\widetilde {I}$, and describe an algorithm to compute an arbitrary number of terms in the Taylor expansion of $I_q$. In addition, we also prove that $I_q$ converges pointwise to $\widetilde I$ as $q\to \infty$. Furthermore, we construct a random perturbation $(a_k)_{k \in \mathbb {N}}$ of the sequence $(2^k)_{k \in \mathbb {N}}$ for which $a_{k+1}/a_k \to 2$ as $k\to \infty$, but for which at the same time $(S_n/n)_{n \in \mathbb {N}}$ satisfies a large deviation principle with the same rate function $\widetilde {I}$ as in the independent case, which is surprisingly different from the rate function $I_2$ one might naïvely expect. We relate this fact to the number of solutions of certain Diophantine equations. Together, these results show that large deviation principles for lacunary trigonometric sums are very sensitive to the arithmetic properties of the sequence $(a_k)_{k\in \mathbb N}$. This is particularly noteworthy since no such arithmetic effects are visible in the central limit theorem or in the law of the iterated logarithm for lacunary trigonometric sums. Our proofs use a combination of tools from probability theory, harmonic analysis, and dynamical systems.References
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Bibliographic Information
- Christoph Aistleitner
- Affiliation: Institute of Analysis and Number Theory, Graz University of Technology, Steyrergasse 30/II, 8010 Graz, Austria
- MR Author ID: 831296
- ORCID: 0000-0002-1460-6164
- Email: aistleitner@math.tugraz.at
- Nina Gantert
- Affiliation: Department of Mathematics, Technical University of Munich, Parkring 11, 85748 Garching, Germany
- MR Author ID: 292404
- ORCID: 0000-0003-0811-3651
- Email: nina.gantert@tum.de
- Zakhar Kabluchko
- Affiliation: Institute for Mathematical Stochastics, University of Münster, Orléans-Ring 10, 48149 Münster, Germany
- MR Author ID: 696619
- ORCID: 0000-0001-8483-3373
- Email: zakhar.kabluchko@uni-muenster.de
- Joscha Prochno
- Affiliation: Faculty of Computer Science and Mathematics, University of Passau, Innstraße 33, 94032 Passau, Germany
- MR Author ID: 997160
- ORCID: 0000-0002-0750-2850
- Email: joscha.prochno@uni-passau.de
- Kavita Ramanan
- Affiliation: Division of Applied Mathematics, Brown University, 182 George Street, Providence, Rhode Island 02912
- MR Author ID: 634978
- Email: kavita_ramanan@brown.edu
- Received by editor(s): November 29, 2021
- Received by editor(s) in revised form: June 28, 2022
- Published electronically: October 14, 2022
- Additional Notes: The first author was supported by the Austrian Science Fund (FWF), projects F-5512, I-3466, I-4945 and Y-901. The third author was supported by the German Research Foundation under Germany’s Excellence Strategy EXC 2044 – 390685587, Mathematics Münster: Dynamics - Geometry - Structure. The fourth author was supported by the Austrian Science Fund (FWF), projects P32405 and the Special Research Program F5508-N26. The fifth author was supported by the National Science Foundation (NSF) Grant DMS-1954351 and the Roland George Dwight Richardson Chair at Brown University and a Simon Guggenheim Fellowship. The authors were also supported by the Oberwolfach Research Institute for Mathematics (MFO), where they initiated their collaboration during the workshop on “New Perspectives and Computational Challenges in High Dimensions” (Workshop ID 2006b) in February 2020.
- © Copyright 2022 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 376 (2023), 507-553
- MSC (2020): Primary 42A55, 60F10, 11L03; Secondary 37A05, 11D45, 11K70
- DOI: https://doi.org/10.1090/tran/8788
- MathSciNet review: 4510117
Dedicated: Dedicated to the memory of Nicole Tomczak-Jaegermann