Roots of random polynomials whose coefficients have logarithmic tails

• Autores: Zakhar Kabluchko, Dmitry Zaporozhets
• Localización: Annals of probability: An official journal of the Institute of Mathematical Statistics, ISSN 0091-1798, Vol. 41, Nº. 5, 2013, págs. 3542-3581
• Idioma: inglés
• DOI: 10.1214/12-AOP764
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• Resumen

  • It has been shown by Ibragimov and Zaporozhets [In Prokhorov and Contemporary Probability Theory (2013) Springer] that the complex roots of a random polynomial Gn(z)=∑nk=0ξkzk with i.i.d. coefficients ξ0,…,ξn concentrate a.s. near the unit circle as n→∞ if and only if Elog+|ξ0|<∞. We study the transition from concentration to deconcentration of roots by considering coefficients with tails behaving like L(log|t|)(log|t|)−α as t→∞, where α≥0, and L is a slowly varying function. Under this assumption, the structure of complex and real roots of Gn is described in terms of the least concave majorant of the Poisson point process on [0,1]×(0,∞) with intensity αv−(α+1)dudv.