On expected number of real zeros of a random hyperbolic polynomial with dependent coefficients

The asymptotic estimate of the expected number of real zeros of the random hyperbolic polynomial of the form fn(t)≡fn(t,ω)=y1(ω)cosht+y2(ω)cosh2t+⋯+yn(ω)coshntfn(t)≡fn(t,ω)=y1(ω)cosht+y2(ω)cosh2t+⋯+yn(ω)coshnt is known if the coefficients y1(ω),y2(ω),…,yn(ω)y1(ω),y2(ω),…,yn(ω) are independent and normally distributed random variables with mean zero and variance one. We have considered here the case when the random coefficients are dependent and proved that the expected number of real zeros of fn(t)fn(t) is (1/π)logn+O(1)(1/π)logn+O(1) if the correlation coefficients between yi(ω)yi(ω) and yj(ω)yj(ω) are ρ|i−j|(0<ρ<1,i≠j)ρ|i−j|(0<ρ<1,i≠j) and the expected number of real zeros is O(1) if the correlation coefficients between yi(ω)yi(ω) and yj(ω)yj(ω) are ρ,0<ρ<1ρ,0<ρ<1.