## Electronic Journal of Probability

### CLT for crossings of random trigonometric polynomials

#### Abstract

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

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

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

https://projecteuclid.org/euclid.ejp/1465064293

Digital Object Identifier
doi:10.1214/EJP.v18-2403

Mathematical Reviews number (MathSciNet)
MR3084654

Zentralblatt MATH identifier
1284.60048

Subjects
Primary: 60G15: Gaussian processes

Rights

#### Citation

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

#### References

• Arcones, Miguel A. Limit theorems for nonlinear functionals of a stationary Gaussian sequence of vectors. Ann. Probab. 22 (1994), no. 4, 2242–2274.
• Azaïs, Jean-Marc; Wschebor, Mario. Level sets and extrema of random processes and fields. John Wiley & Sons, Inc., Hoboken, NJ, 2009. xii+393 pp. ISBN: 978-0-470-40933-6.
• Granville, Andrew; Wigman, Igor. The distribution of the zeros of random trigonometric polynomials. Amer. J. Math. 133 (2011), no. 2, 295–357.
• Kratz, Marie F.; León, José R. Hermite polynomial expansion for non-smooth functionals of stationary Gaussian processes: crossings and extremes. Stochastic Process. Appl. 66 (1997), no. 2, 237–252.
• Kratz, Marie F.; León, José R. Central limit theorems for level functionals of stationary Gaussian processes and fields. J. Theoret. Probab. 14 (2001), no. 3, 639–672.
• Nualart, David. The Malliavin calculus and related topics. Second edition. Probability and its Applications (New York). Springer-Verlag, Berlin, 2006. xiv+382 pp. ISBN: 978-3-540-28328-7; 3-540-28328-5.
• Peccati, Giovanni; Tudor, Ciprian A. Gaussian limits for vector-valued multiple stochastic integrals. Séminaire de Probabilités XXXVIII, 247–262, Lecture Notes in Math., 1857, Springer, Berlin, 2005.