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CLT for crossings of random trigonometric polynomials. (English) Zbl 1284.60048
Summary: 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 \(\frac{\sin}{(t)}/t\); b) the use of Wiener chaos decomposition that explains some singularities that appear in the limit when \(u \neq 0\).

60F05 Central limit and other weak theorems
60G15 Gaussian processes
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