Asymptotic zero distribution of random orthogonal polynomials

Thomas Bloom, Duncan Dauvergne

Research output: Contribution to journalArticle

1 Scopus citations

Abstract

We consider random polynomials of the form where the ξj are i.i.d. nondegenerate 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 fail to converge in probability to the appropriate equilibrium measure when the above condition on the ξj is not satisfied. We also consider random polynomials of the form, 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.

Original languageEnglish (US)
Pages (from-to)3202-3230
Number of pages29
JournalAnnals of Probability
Volume47
Issue number5
DOIs
StatePublished - Sep 1 2019
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Statistics and Probability
  • Statistics, Probability and Uncertainty

Keywords

  • Bernstein-markov property
  • Complex zeros
  • Equilibrium measure
  • Logarithmic potential
  • Orthogonal polynomials
  • Potential theory
  • Random polynomials

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