On the average number of level crossings of a random trigonometric polynomial
type: Item
instance of: scholarly article
title: On the average number of level crossings of a random trigonometric polynomial
zbMATH Open document ID: 0708.60064
author: Kambiz Farahmand
published in: The Annals of Probability
publication date: 1990-01-01T00:00:00Z
review text: This is an interesting paper on the average number of level crossings of random trigonometric polynomials. The author obtains an asymptotic estimate of the average number of real roots of the equation \(\sum^{n}_{j=1}g_ j(w) \cos j\theta =K\) in the interval \(0\leq \theta \leq 2\pi\), the coefficients \(g_ j(w)\) being independent, standard normal random variables. He compares this with his earlier paper on the average number of real roots of a random algebraic equation, Ann. Probab. 14, 702-709 (1986; Zbl 0609.60074), and shows that the number of crossings of the algebraic polynomial of the level K decreases as K increases, while for the trigonometric case this remains fixed, with probability 1 as long as \(K^ 2/n\to 0\) as \(n\to \infty.\) He also shows that the results for \(K=0\) are valid when \(K=o(\sqrt{n})\).
Mathematics Subject Classification ID: 60H99
zbMATH DE Number: 4163853
zbMATH Keywords: average number of level crossings
zbMATH Keywords: random trigonometric polynomials
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aop/1176990751
OpenAlex ID: W2063490452
DOI: 10.1214/AOP/1176990751
description: scientific article; zbMATH DE number 4163853