August 2023 Rates of convergence for the number of zeros of random trigonometric polynomials
Laure Coutin, Liliana Peralta
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Bernoulli 29(3): 1983-2007 (August 2023). DOI: 10.3150/22-BEJ1528


In this paper, we quantify the rate of convergence between the distribution of number of zeros of random trigonometric polynomials (RTP) with i.i.d. centered random coefficients and the number of zeros of a stationary centered Gaussian process G, whose covariance function is given by the sinc function. First, we find the convergence of the RTP towards G in the Wasserstein1 distance, which in turn is a consequence of Donsker Theorem. Then, we use this result to derive the rate of convergence between their respective number of zeros. Since the number of real zeros of the RTP is not a continuous function, we use the Kac-Rice formula to express it as the limit of an integral and, in this way, we approximate it by locally Lipschitz continuous functions.

Funding Statement

The first author was partially supported by ANR MESA.


The second author would like to thank the all members to the Institut de Mathématiques de Toulouse, France for their kind hospitality during her research stay.


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Laure Coutin. Liliana Peralta. "Rates of convergence for the number of zeros of random trigonometric polynomials." Bernoulli 29 (3) 1983 - 2007, August 2023.


Received: 1 February 2021; Published: August 2023
First available in Project Euclid: 27 April 2023

MathSciNet: MR4580904
zbMATH: 07691569
Digital Object Identifier: 10.3150/22-BEJ1528

Keywords: Donsker Theorem , random trigonometric polynomials , Stein method , Wassertein distance


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Vol.29 • No. 3 • August 2023
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