ROOTS OF RANDOM POLYNOMIALS WHOSE COEFFICIENTS HAVE LOGARITHMIC TAILS

• 成果类型: Article
• 署名作者: Kabluchko, Zakhar; Zaporozhets, Dmitry
• 署名单位: Ulm University; Russian Academy of Sciences
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/12-AOP764
• 发表日期: 2013
• 页码: 3542-3581
• 关键词: zeros

摘要

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) = Sigma(n)(k=0) xi(k)z(k) with i.i.d. coefficients xi(0), ... , xi(n) concentrate a.s. near the unit circle as n -> infinity if and only if E log(+)vertical bar xi(0)vertical bar < infinity. We study the transition from concentration to deconcentration of roots by considering coefficients with tails behaving like L(log vertical bar t vertical bar)(log vertical bar t vertical bar)(-alpha) as t -> infinity, where alpha >= 0, 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] x (0, infinity) with intensity alpha v(-(alpha+1)) du dv.