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 Vn(x)= ∑j=0naj cos(jx), x∈(0,2π), with the aj being standard Gaussian i.i.d. random variables, is asymptotically 2n∕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∕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 A2j+1=A2j 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.