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

For random polynomials with i.i.d. (independent and identically distribu-ted) zeros following any common probability distribution $μ$ 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 $μ$ a.s. (almost sure(ly)). This, in particular, completes a recent work of Subramanian on the first order derivative case where $μ$ 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.

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