Variance of real zeros of random orthogonal polynomials

Abstract

We determine the asymptotics for the variance of the number of zeros of random linear combinations of orthogonal polynomials of degree $\leq n$ in subintervals $\left [ a,b\right ] $ of the support of the underlying orthogonality measure $\mu $. We show that, as $n\rightarrow \infty $, this variance is asymptotic to $cn$, for some explicit constant $c>0$.