The Annals of Probability

Roots of random polynomials whose coefficients have logarithmic tails

Zakhar Kabluchko and Dmitry Zaporozhets

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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 $\alpha v^{-(\alpha+1)}\,du\,dv$.

Article information

Ann. Probab., Volume 41, Number 5 (2013), 3542-3581.

First available in Project Euclid: 12 September 2013

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Zentralblatt MATH identifier

Primary: 26C10: Polynomials: location of zeros [See also 12D10, 30C15, 65H05]
Secondary: 30B20: Random power series 60G70: Extreme value theory; extremal processes 60B10: Convergence of probability measures 60F10: Large deviations

Random polynomials distribution of roots weak convergence heavy tails least concave majorant extreme value theory


Kabluchko, Zakhar; Zaporozhets, Dmitry. Roots of random polynomials whose coefficients have logarithmic tails. Ann. Probab. 41 (2013), no. 5, 3542--3581. doi:10.1214/12-AOP764.

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