Asymptotics of the variance of the number of real roots of random trigonometric polynomials
type: Item
instance of: scholarly article
title: Asymptotics of the variance of the number of real roots of random trigonometric polynomials
zbMATH Open document ID: 1272.60030
author: Zhonggen Su
author: Q. M. Shao
published in: Science China. Mathematics
publication date: 2013-01-28T00:00:00Z
review text: The average number of real zeros of polynomial \(T_n(\theta)=\sum_{j=1}^{n} a_j \cos j\theta \) where the coefficients \(\{a_j\}_{j=0}^n\) are a sequence of independent identically distributed Gaussian random variables is well-known. This average number, denoted by \(EN_n\) in the internal \((0, 2\pi)\), is asymptotic to \(\operatorname{E}N_n \sim 2n/\sqrt{3} \), for all sufficiently large \(n\). The problem is more complicated when we look for an asymptotic estimate for the variance of the number of real zeros. An upper limit is obtained by the reviewer in [J. Appl. Math. Stochastic Anal. 10, No. 1, 57--66 (1997; Zbl 0880.60058)]. It was acknowledged, since the upper limit obtained was too large, it would not be applicable to obtain an asymptotic estimate for the variance. The present paper successfully re-examined the problem and, using a new approach, obtained an estimate for the variance of the number of real zeros. This paper should pave the way for a study of the distribution of zeros of the above random trigonometric polynomials \(T_n(\theta)\). The latter distribution is known for polynomials of the form \(\sum_{j=1}^{n} (a_j \cos j\theta +b_j \sin j\theta )\), which has the property of being stationary on \(\theta\) and, therefore, the computations are slightly easier. A summary of the earlier results and list of references on this subject is given in [the reviewer, Topics in random polynomials. Harlow: Addison Wesley Longman (1998; Zbl 0949.60010)].
Mathematics Subject Classification ID: 60G99
Mathematics Subject Classification ID: 60G15
zbMATH DE Number: 6131938
zbMATH Keywords: random polynomials
zbMATH Keywords: number of real zeros
zbMATH Keywords: random trigonometric polynomials
zbMATH Keywords: variance of the number of zeros
reviewed by: Kambiz Farahmand
MaRDI profile type: Publication
full work available at URL: s11425-012-4525-5
OpenAlex ID: W2060276784
cites work: Q5556835
cites work: Random polynomials having few or no real zeros
cites work: On the variance of the number of real zeros of a random trigonometric polynomial
cites work: Q4265272
cites work: The distribution of the zeros of random trigonometric polynomials
cites work: Q5644862
cites work: On the Expected Number of Real Zeros of Random Polynomials I. Coefficients with Zero Means
cites work: On the Expected Number of Real Zeros of Random Polynomials. II. Coefficients With Non-Zero Means
cites work: Q4403533
cites work: On the average number of real roots of a random algebraic equation
cites work: Real Zeros of Random Polynomials
cites work: Real Zeros of Random Polynomials. II
cites work: On the Variance of the Number of Real Roots of Random Polynomials
cites work: On the Distribution of the Number of Real Roots of Random Polynomials
cites work: Mean Number of Real Zeros of a Random Trigonometric Polynomial
DOI: 10.1007/S11425-012-4525-5
Recommended article: The distribution of the zeros of random trigonometric polynomials
Recommended article: Mean Number of Real Zeros of a Random Trigonometric Polynomial
Recommended article: On the average number of real zeros of a random trigonometric polynomial with dependent coefficients. II
Recommended article: Covariance of the number of real zeros of a random trigonometric polynomial
Recommended article: On the number of real zeros of a random trigonometric polynomial: coefficients with non–zero infinite mean
Recommended article: The Average Number of Real Zeros of a Random Trigonometric Polynomial
Recommended article: On the variance of the number of real roots of a random trigonometric polynomial
Recommended article: Q3742445
Recommended article: Complex zeros of trigonometric polynomials with standard normal random coefficients
Recommended article: A result concerning real zeros of a random trigonometric polynomial II
description: scientific article; zbMATH DE number 6131938