Covariance of the number of real zeros of a random trigonometric polynomial
type: Item
instance of: scholarly article
title: Covariance of the number of real zeros of a random trigonometric polynomial
zbMATH Open document ID: 1115.60057
DOI: 10.1155/IJMMS/2006/28492
author: Kambiz Farahmand
author: M. Sambandham
published in: International Journal of Mathematics and Mathematical Sciences
publication date: 2007-06-13T00:00:00Z
full work available at URL: https://eudml.org/doc/53215
review text: Summary: For random coefficients \(a_j\) and \(b_j\) we consider a random trigonometric polynomial defined as \(T_n(\theta)=\sum^n_{j=0}\{a_j \cos j\theta+b_j\sin j \theta\}\). The expected number of real zeros of \(T_n(\theta)\) in the interval \((0,2\pi)\) can be easily obtained. We show that this number is in fact \(n/\sqrt 3\). However the variance of the above number is not known. This note presents a method which leads to the asymptotic value for the covariance of the number of real zeros of the above polynomial in intervals \((0,\pi)\) and \((\pi,2\pi)\). It can be seen that our method in fact remains valid to obtain the result for any two disjoint intervals. The applicability of our method to the classical random trigonometric polynomial, defined as \(P_n(\theta)= \sum^n_{j=0} a_j(\omega)\cos j\theta\), is also discussed. \(T_n (\theta)\) has the advantage on \(P_n(\theta)\) of being stationary, with respect to \(\theta\), for which, therefore, a more advanced method developed could be used to yield the results.
Mathematics Subject Classification ID: 60G99
zbMATH DE Number: 5163987
MaRDI profile type: Publication
cites work: Q4722950
cites work: Q5556835
cites work: The Number of Real Zeros of a Random Trigonometric Polynomial
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cites work: Number of real roots of a random trigonometric polynomial
cites work: On the variance of the number of real zeros of a random trigonometric polynomial
cites work: Q4265272
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OpenAlex ID: W2106518578
description: scientific article; zbMATH DE number 5163987