Real zeros of random trigonometric polynomials with pairwise equal blocks of coefficients

Abstract

It is well known that the expected number of real zeros of a random cosine polynomial $V_n\left(x\right)={\sum }_{j=0}^n{a}_{j}cos\left(jx\right)$, $x\in \left(0,2\pi \right)$, with the $a_j$ being standard Gaussian i.i.d. random variables, is asymptotically $2n∕\sqrt{3}$. On the other hand, some of the previous works on the random cosine polynomials with dependent coefficients show that such polynomials have at least $2n∕\sqrt{3}$ expected real zeros lying in one period. We investigate two classes of random cosine polynomials with pairwise equal blocks of coefficients. First, we prove that a random cosine polynomial with the blocks of coefficients being of a fixed length and satisfying $A_{2j+1}=A_{2j}$ possesses the same expected real zeros as the classical case. Afterwards, we study a case containing only two equal blocks of coefficients, and show that in this case significantly more real zeros should be expected compared with those of the classical case.