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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 $$u \neq 0$$.

##### MSC:
 60F05 Central limit and other weak theorems 60G15 Gaussian processes
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