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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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