2018/01/30 by Bloom, Thomas, Dauvergne, Duncan
#30C15 #32U35 #42C05 #60B10 #60G57 #Complex Variables (math.CV) #FOS: Mathematics #Probability (math.PR)
paper · doi:10.48550/arxiv.1801.10125
We consider random polynomials of the form Hn(z)=∑j=0nξjqj(z) where the \ξj\ are i.i.d non-degenerate complex random variables, and the \qj(z)\ are orthonormal polynomials with respect to a compactly supported measure τ satisfying the Bernstein-Markov property on a regular compact set K ⊂ ℂ. We show that if ℙ(|ξ0|>e|z|)=o(|z|-1), then the normalized counting measure of the zeros of Hn converges weakly in probability to the equilibrium measure of K. This is the best possible result, in the sense that the roots of Gn(z)=∑j=0nξjzj fail to converge in probability to the appropriate equilibrium measure when the above condition on the ξj is not satisfied. In addition, we give a multivariable version of this result. We also consider random polynomials of the form ∑k=0nξkfn,kzk, where the coefficients fn,k are complex constants satisfying certain conditions, and the random variables \ξk\ satisfy 𝔼 log(1 + |ξ0|) < ∞. In this case, we establish almost sure convergence of the normalized counting measure of the zeros to an appropriate limiting measure. Again, this is the best possible result in the same sense as above.