Real zeros of random cosine polynomials with palindromic blocks of coefficients (Q822642)

Classical results on a random cosine polynomial \(V_n=\sum_{j=0}^{\infty} a_j cos(jx),\) \(x\in (0, 2\pi)\) state that if the coefficients \(a_j\) are independent and identically distributed (i.i.d.) real-valued standard Gaussian 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 \((0, 2\pi).\) This paper is one from the series of the author's works devoted to 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. \par In the present paper the author investigates how many real zeros a random cosine polynomial \(V_n\) with identically and normally distributed coefficients possesses if the coefficients are sorted in palindromic blocks of a fixed length \(l.\) More precisely, assume that \(l\in \mathbb{N}\) is 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 of the length \(l\) in the following fashion: \( (a_0, a_1, \ldots, a_n)=\cup_{j=0}^{2m-1} A_{j} \cup \tilde{A}, \) where \(A_j=(a_{ l j}, a_{ l j+1}, \ldots, a_{ l j+l-1})\) if \( j=0, \ldots, m-1;\) \(A_j=(a_{ l j+r}, a_{ l j+1+r}, \ldots, a_{ l j+l-1+r})\) if \( j=m, \ldots, 2m-1;\) \(\tilde{A}=(a_{ l m}, a_{ l m+1}, \ldots, a_{ l m+r-1})\) if \( 1\leq r\leq 2l-1,\) and \(\tilde{A}=\emptyset\) if \(r=0.\) The author proves that if \(\cup_{j=0}^{m-1} A_{2j} \cup \tilde{A}\) is a family of i.i.d. random variables with Gaussian distribution \( \mathcal{N}(0, \sigma^2),\) and \(a_{l(2m-1-j)+r+k}=a_{l j+k}\) for \(j=0,1, \ldots, m-1,\) \(k =0,1, \ldots, l-1,\) (i.e. \(A_{2m-1-j}=A_j\)), then \(\mathbb{E}[N_n(0, 2\pi)] = \frac{2n}{\sqrt{3}}K_l+\mathcal{O}(n^{3/4}), \) as \(n \to \infty,\) where the constant \(K_l>1\) depends only on \(l\) and can be explicitly represented by a double integral formula. That is in this case slightly more real zeros should be expected compared with the classical case of random cosine polynomials with i.i.d. coefficients.