### Abstract

We consider random polynomials of the form where the ξ_{j} are i.i.d. nondegenerate complex random variables, and the q_{j}(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(|ξ_{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 fail to converge in probability to the appropriate equilibrium measure when the above condition on the ξ_{j} is not satisfied. We also consider random polynomials of the form, where the coefficients f_{n, k} are complex constants satisfying certain conditions, and the random variables ξ_{k} satisfy E log(1+|ξ_{0}|) <∞. 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 language | English (US) |
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Pages (from-to) | 3202-3230 |

Number of pages | 29 |

Journal | Annals of Probability |

Volume | 47 |

Issue number | 5 |

DOIs | |

State | Published - Sep 1 2019 |

Externally published | Yes |

### 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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## Cite this

*Annals of Probability*,

*47*(5), 3202-3230. https://doi.org/10.1214/19-AOP1337