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 $\le n$ in subintervals $\left[a,b\right]$ of the support of the underlying orthogonality measure $\mu$. We show that, as $n\to \infty$, this variance is asymptotic to $cn$, for some explicit constant $c>0$.

26 pages; to appear in J. Math. Anal. Appl