Electronic Journal of Probability

CLT for crossings of random trigonometric polynomials

Jean-Marc Azaïs and José León

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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 $u \neq 0$.

Article information

Electron. J. Probab., Volume 18 (2013), paper no. 68, 17 pp.

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G15: Gaussian processes

Crossings of random trigonometric polynomials Rice formula Chaos expansion

This work is licensed under a Creative Commons Attribution 3.0 License.


Azaïs, Jean-Marc; León, José. CLT for crossings of random trigonometric polynomials. Electron. J. Probab. 18 (2013), paper no. 68, 17 pp. doi:10.1214/EJP.v18-2403. https://projecteuclid.org/euclid.ejp/1465064293

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