The Expected Number of Real Zeros of Algebraic Polynomials with Dependent and Non-Identical Random Coefficients

Abstract

This article provides an asymptotic formula for the expected number of real zeros of a polynomial of the form for large n. The coefficients are assumed to be a sequence of dependent normally distributed random variables with E(a j (ω)) = 0, var(a j (ω)) = 1 and cov(a i (ω), a j (ω)) = ρ, 0 < ρ <1. We show that for the above dependent case this expected number is half that for the independent case. This behavior is similar to that of classical random algebraic polynomials.