Zeros of a growing number of derivatives of random polynomials with independent roots

Abstract

Let X 1 , X 2 , … X_1,X_2,\ldots be independent and identically distributed random variables in C {\mathbb {C}} chosen from a probability measure μ \mu and define the random polynomial P n ( z ) = ( z − X 1 ) … ( z − X n ) . \begin{align*} P_n(z)=(z-X_1)\ldots (z-X_n)\,. \end{align*} We show that for any sequence k = k ( n ) k = k(n) satisfying k ≤ log ⁡ n / ( 5 log ⁡ log ⁡ n ) k \leq \log n / (5 \log \log n) , the zeros of the k k th derivative of P n P_n are asymptotically distributed according to the same measure μ \mu . This extends work of Kabluchko, which proved the k = 1 k = 1 case, as well as Byun, Lee and Reddy [Trans. Amer. Math. Soc. 375, pp. 6311–6335] who proved the fixed k k case.