Expected number of real zeros of random hyperbolic polynomial
type: Item
instance of: scholarly article
title: Expected number of real zeros of random hyperbolic polynomial
zbMATH Open document ID: 1113.60069
author: Mina Ketan Mahanti
published in: Statistics \& Probability Letters
publication date: 2007-03-02T00:00:00Z
review text: There are many known results concerning the mathematical expectation of the number of real zeros of the random algebraic polynomial \(\sum_{k=0}^{n}g_k t^k\) where \(\{g_k\}_{k=0}^n\) is a sequence of independent identically distributed normal standard random variables. However, in comparison, very little is known about the polynomials in the form of \(f_n(t)= \sum_{k=0}^n g_k \cosh kt\). The analysis involved with the latter polynomial becomes complicated and therefore less information is known. The asymptotic value for the expected number of real zeros of \(f_n(t)\) denoted by \(EN_n\) is obtained, by various authors, and known to be \((1/\pi)\log n\). The error term involved in the calculation of these asymptotic values, however, is large and generally is an increasing function of \(n\). This paper shows that this error term in fact is \(O(1)\). The author gives a very detailed estimate of the terms involved in calculation of \(EN_n\) and proves that \(EN_n\sim (1/\pi)\log n+0.038...+o(1)\). The result is obtained by dividing the intervals \((-\infty, \infty)\) to several subinterval and using different methods in each. These methods should be of interest to those who would like to generalize the known results for \(f_n(t)\) to other cases already developed for the random algebraic polynomials. For earlier results on the subject see \textit{A. T. Bharucha-Reid} and \textit{M. Sambandham} [``Random polynomials, probability and mathematical statistics'' (1986; Zbl 0615.60058)] and the reviewer [``Topics in random polynomials'' (1998; Zbl 0949.60010)] and their references.
reviewed by: Kambiz Farahmand
Mathematics Subject Classification ID: 60H99
Mathematics Subject Classification ID: 26C99
zbMATH DE Number: 5130369
zbMATH Keywords: Random polynomials
zbMATH Keywords: Hyperbolic polynomials
zbMATH Keywords: Number of real zeros
zbMATH Keywords: Jensen's theorem
MaRDI profile type: Publication
full work available at URL: j.spl.2004.06.017
OpenAlex ID: W1992573077
cites work: Q4722950
cites work: Q5556835
cites work: Level Crossings of a Random Polynomial with Hyperbolic Elements
cites work: Q3671491
cites work: Q3259971
cites work: AN EXAMPLE IN THE THEORY OF THE SPECTRUM OF A FUNCTION
cites work: An Asymptotic Expansion for the Expected Number of Real Zeros of a Random Polynomial
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DOI: 10.1016/J.SPL.2004.06.017
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description: scientific article; zbMATH DE number 5130369