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September 2013 Roots of random polynomials whose coefficients have logarithmic tails
Zakhar Kabluchko, Dmitry Zaporozhets
Ann. Probab. 41(5): 3542-3581 (September 2013). DOI: 10.1214/12-AOP764


It has been shown by Ibragimov and Zaporozhets [In Prokhorov and Contemporary Probability Theory (2013) Springer] that the complex roots of a random polynomial $G_{n}(z)=\sum_{k=0}^{n}\xi_{k}z^{k}$ with i.i.d. coefficients $\xi_{0},\ldots,\xi_{n}$ concentrate a.s. near the unit circle as $n\to\infty$ if and only if ${\mathbb{E}\log_{+}}|\xi_{0}|<\infty$. We study the transition from concentration to deconcentration of roots by considering coefficients with tails behaving like $L({\log}|t|)({\log}|t|)^{-\alpha}$ as $t\to\infty$, where $\alpha\geq0$, and $L$ is a slowly varying function. Under this assumption, the structure of complex and real roots of $G_{n}$ is described in terms of the least concave majorant of the Poisson point process on $[0,1]\times (0,\infty)$ with intensity $\alpha v^{-(\alpha+1)}\,du\,dv$.


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Zakhar Kabluchko. Dmitry Zaporozhets. "Roots of random polynomials whose coefficients have logarithmic tails." Ann. Probab. 41 (5) 3542 - 3581, September 2013.


Published: September 2013
First available in Project Euclid: 12 September 2013

zbMATH: 1364.26019
MathSciNet: MR3127891
Digital Object Identifier: 10.1214/12-AOP764

Primary: 26C10
Secondary: 30B20 , 60B10 , 60F10 , 60G70

Keywords: distribution of roots , Extreme value theory , heavy tails , least concave majorant , random polynomials , weak convergence

Rights: Copyright © 2013 Institute of Mathematical Statistics


Vol.41 • No. 5 • September 2013
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