an:07261874 Zbl 1461.30027 Pirhadi, Ali Real zeros of random trigonometric polynomials with pairwise equal blocks of coefficients EN Rocky Mt. J. Math. 50, No. 4, 1451-1471 (2020). 0035-7596 2020
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30C15 30C99 12D10 26C10 random trigonometric polynomials; dependent coefficients; expected number of real zeros The paper deals with a random trigonometric polynomial $$V_n=\sum_{j=0}^{\infty} a_j \cos(jx),$$ $$x\in (0, 2\pi).$$ Classical results state that if the coefficients $$a_j$$ are standard Gaussian i.i.d. random variables, then the expected value $$\mathbb{E}[N_n(0, 2\pi)] \sim \frac{2n}{\sqrt{3}}$$ as $$n\to \infty,$$ where $$N_n(0, 2\pi)$$ denotes the number of real zeros of $$V_n$$ in $$(0, 2\pi).$$ The main focus of the paper is the question on how many real zeros, compared with the case of independent coefficients, should be expected if a certain restriction is imposed upon the coefficients. Let $$l\in \mathbb{N}$$ be a fixed number and $$n=2l m-1+r,$$ where $$m\in\mathbb{N},$$ and $$r \in \{0,1,\ldots, 2l-1\}.$$ The coefficients $$A=(a_j)_{j=0}^n$$ are divided into $$2m$$ blocks $$A_j=(a_{ l j}, a_{ l j+1}, \ldots, a_{ l j+l-1}), \ j=0, \ldots, 2m-1,$$ of the length $$l,$$ and the remaining coefficients $$\tilde{A}.$$ Assume that $$\bigcup_{j=0}^{m-1} A_{2j} \cup \tilde{A}$$ is a family of i.i.d. random variables with Gaussian distribution $$\mathcal{N}(0, \sigma^2),$$ and $$a_{l(2j+1)+k}=a_{2lj+k}$$ for any $$j=0, \ldots, m-1$$ and $$k=0, \ldots, l-1,$$ that is, $$A_{2j+1}=A_{2j}$$. The author proves that under these assumptions $$\mathbb{E}[N_n(0, 2\pi)] = \frac{2n}{\sqrt{3}}+\mathcal{O}(n^{4/5})$$, as $$n \to \infty.$$ In the case of only two equal blocks of coefficients, the author obtains the following asymptotic formula $$\mathbb{E}[N_n(0, 2\pi)] = \left(\frac{1}{2}+\frac{\sqrt{13}}{2\sqrt{3}}\right)n+\mathcal{O}(n^{4/5}),$$ as $$n \to \infty,$$ that is, in this case significantly more real zeros should be expected compared with those of the classical case. Olga M. Katkova (Boston)