REAL ZEROS OF RANDOM TRIGONOMETRIC POLYNOMIALS WITH PAIRWISE EQUAL BLOCKS OF COEFFICIENTS

摘要

It is well known that the expected number of real zeros of a random cosine polynomial V-n(x) = Sigma(n)(j=0) a(j) cos(jx), x is an element of (0, 2 pi), with the a(j) being standard Gaussian i.i.d. random variables, is asymptotically 2n/root 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/root 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+i) = 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.