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 $u \neq 0$.