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2013 CLT for crossings of random trigonometric polynomials
Jean-Marc Azaïs, José León
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Electron. J. Probab. 18: 1-17 (2013). DOI: 10.1214/EJP.v18-2403


We establish a central limit theorem for the number of roots of the equation $X_N(t) =u$ when $X_N(t)$ is a Gaussian trigonometric polynomial of degree $N$. The case $u=0$ was studied by Granville and Wigman. We show that for some size of the considered interval, the asymptotic behavior is different depending on whether $u$ vanishes or not. Our mains tools are: a) a chaining argument with the stationary Gaussain process with covariance $\sin(t)/t$, b) the use of Wiener chaos decomposition that explains some singularities that appear in the limit when $u \neq 0$.


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Jean-Marc Azaïs. José León. "CLT for crossings of random trigonometric polynomials." Electron. J. Probab. 18 1 - 17, 2013.


Accepted: 18 July 2013; Published: 2013
First available in Project Euclid: 4 June 2016

zbMATH: 1284.60048
MathSciNet: MR3084654
Digital Object Identifier: 10.1214/EJP.v18-2403

Primary: 60G15

Keywords: chaos expansion , Crossings of random trigonometric polynomials , Rice formula

Vol.18 • 2013
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