Higher Order, Polar and Sz.-Nagy's Generalized Derivatives of Random Polynomials with Independent and Identically Distributed Zeros on the Unit Circle

TL;DR

This paper proves that the empirical measures of zeros of higher order derivatives, including polar and Sz.-Nagy's generalized derivatives, of random polynomials with i.i.d. zeros on the unit circle converge almost surely to the original distribution, extending previous results.

Contribution

It extends the almost sure weak convergence results of zeros to higher order, polar, and Sz.-Nagy's derivatives of random polynomials with i.i.d. zeros on the unit circle.

Findings

Empirical measures of zeros of higher order derivatives converge almost surely to the original distribution.

Results include polar and Sz.-Nagy's generalized derivatives under mild conditions.

Completes and extends previous work on the first order derivative case.

Abstract

For random polynomials with i.i.d. (independent and identically distribu-ted) zeros following any common probability distribution $\mu$ with support contained in the unit circle, the empirical measures of the zeros of their first and higher order derivatives will be proved to converge weakly to $\mu$ a.s. (almost sure(ly)). This, in particular, completes a recent work of Subramanian on the first order derivative case where $\mu$ was assumed to be non-uniform. The same a.s. weak convergence will also be shown for polar and Sz.-Nagy's generalized derivatives, on some mild conditions.