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
j
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)