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2023 Fluctuations in Salem–Zygmund almost sure Central Limit Theorem
Jürgen Angst, Guillaume Poly
Author Affiliations +
Electron. J. Probab. 28: 1-40 (2023). DOI: 10.1214/23-EJP931


We consider the following family of random trigonometric polynomials of the form Sn(θ):=1nk=1nakcos(kθ)+bksin(kθ), where the variables {ak,bk}k1 are i.i.d. and symmetric, defined on a common probability space (Ω,F,P). In the seminal paper [SZ54], Salem and Zygmund proved that for Rademacher coefficients, Palmost surely and for all tR


To the best of our knowledge, the natural question of the fluctuations in the above almost sure limit has not been tackled so far and is precisely the object of this article. Namely, for general i.i.d. symmetric random coefficients having a finite sixth-moment and for a large class of continuous test functions ϕ we prove that


where the limit variance is given by Σϕ2:=σϕ2+c2(ϕ)22E[a14]3. Here, the constant σϕ2 is explicit and corresponds to the limit variance in the case of Gaussian coefficients and c2(ϕ) is the coefficient of order 2 in the decomposition of ϕ in the Hermite polynomial basis. It thus turns out that the fluctuations are Gaussian and not universal since they involve the kurtosis of the random coefficients.

The method we develop here is robust and allows to use Wiener chaos expansions for proving CLTs for functionals of general, non-necessarily Gaussian, random fields. It combines contributions from the Malliavin–Stein method, noise stability results, and algebraic considerations around Newton sums and the Newton–Girard formula.

Funding Statement

This work was supported by the ANR grant UNIRANDOM, ANR-17-CE40-0008.


Download Citation

Jürgen Angst. Guillaume Poly. "Fluctuations in Salem–Zygmund almost sure Central Limit Theorem." Electron. J. Probab. 28 1 - 40, 2023.


Received: 10 February 2022; Accepted: 7 March 2023; Published: 2023
First available in Project Euclid: 20 March 2023

arXiv: 2111.12571
Digital Object Identifier: 10.1214/23-EJP931

Primary: 26C10
Secondary: 30C15 , 42A05 , 60F17 , 60G55

Keywords: almost sure CLT , Noise sensitivity , random trigonometric polynomials , Universality

Vol.28 • 2023
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