On the number of real zeros of a random trigonometric polynomial

Source abstract

For the random trigonometric polynomial $$∑ n = 1 N g n ( t ) cos ⁡ n θ , \sum \limits _{n = 1}^N {{g_n}(t)\cos n\theta ,}$$ where g n ( t ) , 0 ⩽ t ⩽ 1 {g_n}(t),0 \leqslant t \leqslant 1 , are dependent normal random variables with mean zero, variance one and joint density function $$| M | 1 / 2 ( 2 π ) − N / 2 exp ⁡ [ − ( 1 / 2 ) a ¯ ′ M a ¯ ] |M{|^{1/2}}{(2\pi )^{ - N/2}}\exp [ - (1/2)\bar a’M\bar a]$$ where M − 1 {M^{ - 1}} is the moment matrix with ρ i j = ρ , 0 > ρ > 1 , i ≠ j , i , j = 1 , 2 , … , N {\rho _{ij}} = \rho ,0 > \rho > 1,i \ne j,i,j = 1,2, \ldots ,N and a ¯ \bar a is the column vector, we estimate the probable number of zeros.