{
  "id": "1110.2585",
  "version": "v2",
  "published": "2011-10-12T07:29:39.000Z",
  "updated": "2013-10-21T06:25:16.000Z",
  "title": "Roots of random polynomials whose coefficients have logarithmic tails",
  "authors": [
    "Zakhar Kabluchko",
    "Dmitry Zaporozhets"
  ],
  "comment": "Published in at http://dx.doi.org/10.1214/12-AOP764 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)",
  "journal": "Annals of Probability 2013, Vol. 41, No. 5, 3542-3581",
  "doi": "10.1214/12-AOP764",
  "categories": [
    "math.PR",
    "math.CV"
  ],
  "abstract": "It has been shown by Ibragimov and Zaporozhets [In Prokhorov and Contemporary Probability Theory (2013) Springer] that the complex roots of a random polynomial $G_n(z)=\\sum_{k=0}^n\\xi_kz^k$ 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 $\\alpha v^{-(\\alpha+1)}\\,du\\,dv$.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2013-10-21T06:25:16.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "random polynomial",
      "logarithmic tails",
      "coefficients",
      "contemporary probability theory",
      "poisson point process"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1110.2585K"
    }
  }
}