Exceedence Measure of Classes of Algebraic Polynomials

Abstract

There is both mathematical and physical interest in the behaviour of the polynomial of the form $a_0 + a_1 (_{\text{1}}^n {\kern 1pt} )^{1/2} x + a_2 (_{\text{2}}^n {\kern 1pt} )^{1/2} x^2 + \cdots + a_n (_n^n {\kern 1pt} )^{1/2} x^n$. The coefficients a _j , j = 0,...,n are assumed to be independent normally distributed random variables with mean zero and variance σ ². 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.