Almost sure behavior of the zeros of iterated derivatives of random polynomials

Let Z1,Z2,… be independent and identically distributed complex random variables with common distribution μ and set Pn(z):=(z−Z1)⋯(z−Zn). Recently, Angst, Malicet and Poly proved that the critical points of Pn converge in an almost-sure sense to the measure μ as n tends to infinity, thereby confirming a conjecture of Cheung-Ng-Yam and Kabluchko. In this short note, we prove for any fixed k∈N, the empirical measure of zeros of the kth derivative of Pn converges to μ in the almost sure sense, as conjectured by Angst-Malicet-Poly.