A necessary and sufficient condition for convergence of the zeros of random polynomials

Abstract

Consider random polynomials of the form Gn=∑i=0nξipi, where the ξi are i.i.d. non-degenerate complex random variables, and {pi} is a sequence of orthonormal polynomials with respect to a regular measure τ supported on a compact set K. We show that the zero measure of Gn converges weakly almost surely to the equilibrium measure of K if and only if Elog⁡(1+|ξ0|)<∞. This generalizes the corresponding result of Ibragimov and Zaporozhets in the case when pi(z)=zi. We also show that the zero measure of Gn converges weakly in probability to the equilibrium measure of K if and only if P(|ξ0|>en)=o(n−1).Our proofs rely on results from small ball probability and exploit the structure of general orthogonal polynomials. Our methods also work for sequences of asymptotically minimal polynomials in Lp(τ), where p∈(0,∞]. In particular, sequences of Lp-minimal polynomials and (normalized) Faber and Fekete polynomials fall into this class.