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Exceedence Measure of Classes of Algebraic Polynomials

Authors
  • Farahmand, K.1
  • 1 University of Ulster, Department of Mathematics, Jordanstown, Co. Antrim, BT37 0QB, United Kingdom , Jordanstown, Co. Antrim
Type
Published Article
Journal
Journal of Theoretical Probability
Publisher
Kluwer Academic Publishers-Plenum Publishers
Publication Date
Apr 01, 2003
Volume
16
Issue
2
Pages
419–426
Identifiers
DOI: 10.1023/A:1023526828571
Source
Springer Nature
Keywords
License
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Abstract

There is both mathematical and physical interest in the behaviour of the polynomial of the form \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$a_0 + a_1 (_{\text{1}}^n {\kern 1pt} )^{1/2} x + a_2 (_{\text{2}}^n {\kern 1pt} )^{1/2} x^2 + \cdots + a_n (_n^n {\kern 1pt} )^{1/2} x^n $$ \end{document}. The coefficients aj, j = 0,...,n are assumed to be independent normally distributed random variables with mean zero and variance σ2. In this paper by using the motion of exceedence measure for stochastic processes, for n large, we derive an asymptotic estimate for the expected area of the curve representing the above polynomial cut off by the x-axis. We show that our method can be used to obtain results for similar random polynomials.

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