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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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##### References:
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