On the lower bound of the number of real roots of a random algebraic equation
type: Item
instance of: scholarly article
title: On the lower bound of the number of real roots of a random algebraic equation
zbMATH Open document ID: 0870.60049
DOI: 10.1016/0167-7152(95)00214-6
author: Takashi Uno
published in: Statistics \& Probability Letters
publication date: 1997-06-02T00:00:00Z
review text: Let \(F_n(x,\omega)\equiv F(x)=\sum^n_{\nu=0} a_\nu(\omega)b_\nu x^\nu\), where \(\{a_\nu(\omega)\}^n_{\nu=0}\) is a sequence of normally distributed random variables defined on a probability space \((\Omega,{\mathcal A},\text{Pr})\) and \(\{b_\nu\}^n_{\nu=0}\) is a sequence of positive numbers. There are many results concerning the upper and lower bounds of the number of real zeros of \(F(x)\) under some assumption for the distribution of the coefficients \(a_\nu\) and some conditions for \(b_\nu\). The present paper obtains a lower bound for the number of real zeros for a case of dependent \(a_\nu\). It is assumed that the constants \(b_\nu\) satisfy the relation \(k_n/t_n= o(\log n)\), where \(k_n=\max_{0\leq\nu\leq n}b_\nu\) and \(t_n=\min_{0\leq\nu\leq n}b_\nu\). It is shown that for \(n\) large and \(C\) and \(C'\) positive constants, outside a set of measure at most \(C'\log((k_n/t_n)\log n)/\log n\) there are at least \(C\log n/\{\log((k_n/t_n)\log n)\}\) real zeros for \(F(x)\). The proof is similar to the earlier works of \textit{G. Samal} [Proc. Camb. Philos. Soc. 58, 433-442 (1962; Zbl 0113.34201)] and \textit{G. Samal} and \textit{M. N. Mishra} [Proc. Am. Math. Soc. 33, 523-528 (1972; Zbl 0214.16102) and ibid. 39, 184-189 (1973; Zbl 0274.60044)]. However, the details differ to accommodate the constants \(b_\nu\) and the dependent coefficients.
Mathematics Subject Classification ID: 60G99
zbMATH DE Number: 954944
zbMATH Keywords: random algebraic equations
zbMATH Keywords: dependent normal distribution
zbMATH Keywords: real roots
reviewed by: Kambiz Farahmand
MaRDI profile type: Publication
full work available at URL: 0167-7152(95)00214-6
OpenAlex ID: W2071065864
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description: scientific article; zbMATH DE number 954944