Sharp crossings of a non-stationary stochastic process and its application to random polynomials

Abstract

In this paper we provide the expected number of zero up-crossings with slope greater than udown-crossing with slope less than —uof a Gaussian process ξ(t) Where uis any positive constant. Promoted by graphical interpretation, we define hese crossings as u—sharp. Then the expected number of such crossings of a random lgebraic polynomial of the form with normally distributed coefficients follows from this result. It is Shown that for any bounded uthe expected number of u-sharp crossings is asymptotically equal to 0-sharp crossings while for u→ ∞ as n→ ∞ such that (u 2/3/n)→0 the expected number in he interval (-1,1) asymptotically remains as (1/π) log nand, outside this interval, asymtotically reduces to