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September 2019 Asymptotic zero distribution of random orthogonal polynomials
Thomas Bloom, Duncan Dauvergne
Ann. Probab. 47(5): 3202-3230 (September 2019). DOI: 10.1214/19-AOP1337

Abstract

We consider random polynomials of the form $H_{n}(z)=\sum_{j=0}^{n}\xi_{j}q_{j}(z)$ where the $\{\xi_{j}\}$ are i.i.d. nondegenerate complex random variables, and the $\{q_{j}(z)\}$ are orthonormal polynomials with respect to a compactly supported measure $\tau $ satisfying the Bernstein–Markov property on a regular compact set $K\subset\mathbb{C}$. We show that if $\mathbb{P}(|\xi_{0}|>e^{|z|})=o(|z|^{-1})$, then the normalized counting measure of the zeros of $H_{n}$ converges weakly in probability to the equilibrium measure of $K$. This is the best possible result, in the sense that the roots of $G_{n}(z)=\sum_{j=0}^{n}\xi_{j}z^{j}$ fail to converge in probability to the appropriate equilibrium measure when the above condition on the $\xi_{j}$ is not satisfied.

We also consider random polynomials of the form $\sum_{k=0}^{n}\xi_{k}f_{n,k}z^{k}$, where the coefficients $f_{n,k}$ are complex constants satisfying certain conditions, and the random variables $\{\xi_{k}\}$ satisfy $\mathbb{E}\log (1+|\xi_{0}|)<\infty $. 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.

Citation

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Thomas Bloom. Duncan Dauvergne. "Asymptotic zero distribution of random orthogonal polynomials." Ann. Probab. 47 (5) 3202 - 3230, September 2019. https://doi.org/10.1214/19-AOP1337

Information

Received: 1 March 2018; Revised: 1 January 2019; Published: September 2019
First available in Project Euclid: 22 October 2019

zbMATH: 07145315
MathSciNet: MR4021249
Digital Object Identifier: 10.1214/19-AOP1337

Subjects:
Primary: 30C15
Secondary: 42C05, 60B10, 60G57

Rights: Copyright © 2019 Institute of Mathematical Statistics

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Vol.47 • No. 5 • September 2019
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