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 f n ( t ) ≡ f n ( t , ω ) = y 1 ( ω ) cosh t + y 2 ( ω ) cosh 2 t + ⋯ + y n ( ω ) cosh n t is known if the coefficients y 1 ( ω ) , y 2 ( ω ) , … , y n ( 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 f n ( t ) is ( 1 / π ) log n + O ( 1 ) if the correlation coefficients between y i ( ω ) and y j ( ω ) are ρ | i − j | ( 0 < ρ < 1 , i ≠ j ) and the expected number of real zeros is O(1) if the correlation coefficients between y i ( ω ) and y j ( ω ) are ρ , 0 < ρ < 1 .