## Critical points of random polynomials with independent identically distributed roots

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- by Zakhar Kabluchko
- Proc. Amer. Math. Soc.
**143**(2015), 695-702 - DOI: https://doi.org/10.1090/S0002-9939-2014-12258-1
- Published electronically: September 19, 2014
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## Abstract:

Let $X_1,X_2,\ldots$ be independent identically distributed random variables with values in $\mathbb {C}$. Denote by $\mu$ the probability distribution of $X_1$. Consider a random polynomial $P_n(z)=(z-X_1)\ldots (z-X_n)$. We prove a conjecture of Pemantle and Rivin [in: I. Kotsireas and E. V. Zima, eds.,*Advances in Combinatorics*, Waterloo Workshop in Computer Algebra, $2011$] that the empirical measure \[ \mu _n:=\frac 1{n-1}\sum _{P_n’(z)=0} \delta _z\] counting the complex zeros of the derivative $P_n’$ converges in probability to $\mu$, as $n\to \infty$.

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## Bibliographic Information

**Zakhar Kabluchko**- Affiliation: Institute of Stochastics, Ulm University, Helmholtzstr. 18, 89069 Ulm, Germany
- MR Author ID: 696619
- ORCID: 0000-0001-8483-3373
- Email: zakhar.kabluchko@uni-ulm.de
- Received by editor(s): July 4, 2012
- Received by editor(s) in revised form: February 12, 2013, and May 3, 2013
- Published electronically: September 19, 2014
- Communicated by: David Levin
- © Copyright 2014
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc.
**143**(2015), 695-702 - MSC (2010): Primary 30C15; Secondary 60G57, 60B10
- DOI: https://doi.org/10.1090/S0002-9939-2014-12258-1
- MathSciNet review: 3283656