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New asymptotics for the mean number of zeros of random trigonometric polynomials with strongly dependent Gaussian coefficients. (English) Zbl 1445.60052
Summary: We consider random trigonometric polynomials of the form \[f_n(t):=\frac{1} {\sqrt{n}} \sum_{k=1}^na_k \cos (k t)+b_k \sin (k t),\] where \((a_k)_{k\geq 1}\) and \((b_k)_{k\geq 1}\) are two independent stationary Gaussian processes with the same correlation function \(\rho : k \mapsto \cos (k\alpha)\), with \(\alpha \geq 0\). We show that the asymptotics of the expected number of real zeros differ from the universal one \(\frac{2} {\sqrt{3}}\), holding in the case of independent or weakly dependent coefficients. More precisely, for all \(\varepsilon >0\), for all \(\ell \in (\sqrt{2} ,2]\), there exists \(\alpha \geq 0\) and \(n\geq 1\) large enough such that \[\left |\frac{\mathbb{E} \left [\mathcal{N}(f_n,[0,2\pi ])\right]} {n}-\ell \right |\leq \varepsilon ,\] where \(\mathcal{N} (f_n,[0,2\pi ])\) denotes the number of real zeros of the function \(f_n\) in the interval \([0,2\pi]\). Therefore, this result provides the first example where the expected number of real zeros does not converge as \(n\) goes to infinity by exhibiting a whole range of possible subsequential limits ranging from \(\sqrt{2}\) to 2.
MSC:
60H99 Stochastic analysis
60G99 Stochastic processes
26C10 Real polynomials: location of zeros
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