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Expected number of real roots of random trigonometric polynomials. (English) Zbl 1377.60063
Summary: We investigate the asymptotics of the expected number of real roots of random trigonometric polynomials $X_n(t)=u+\frac{1}{\sqrt{n}}\sum_{k=1}^{n}(A_k\cos(kt)+B_k\sin(kt)),\quad t\in [0,2\pi],\quad u\in \mathbb{R}$ whose coefficients $$A_k$$, $$B_k$$, $$k\in \mathbb{N}$$, are independent identically distributed random variables with zero mean and unit variance. If $$N_n[a,b]$$ denotes the number of real roots of $$X_n$$ in an interval $$[a,b]\subseteq [0,2\pi]$$, we prove that $\lim_{n\to \infty}\frac{\mathbb{E}N_n[a,b]}{n}=\frac{b-a}{\pi\sqrt{3}}\exp \left(-\frac{u^2}{2}\right)$ .

##### MSC:
 60G99 Stochastic processes 60F05 Central limit and other weak theorems 42A05 Trigonometric polynomials, inequalities, extremal problems
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##### References:
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