On the average number of sharp crossings of certain Gaussian random polynomials

on the average number of sharp crossings of certain gaussian random polynomials

On the Average Number of Sharp Crossings of Certain Gaussian Random Polynomials

Let Qn(x) = ∑n i=0 Aix i be a random algebraic polynomial where the coefficients A0, A1, · · · form a sequence of centered Gaussian random variables. Moreover, assume that the increments ∆j = Aj−Aj−1, j = 0, 1, 2, · · · are independent, assuming A−1 = 0. The coefficients can be considered as n consecutive observations of a Brownian motion. We obtain the asymptotic behaviour of the expected numb...

Expected Number of Slope Crossings of Certain Gaussian Random Polynomials

Let Qn(x) = ∑n i=0 Aix i be a random polynomial where the coefficients A0, A1, · · · form a sequence of centered Gaussian random variables. Moreover, assume that the increments ∆j = Aj − Aj−1, j = 0, 1, 2, · · · are independent, assuming A−1 = 0. The coefficients can be considered as n consecutive observations of a Brownian motion. We study the number of times that such a random polynomial cros...

Expected Number of Local Maxima of Some Gaussian Random Polnomials

On Classifications of Random Polynomials

 Let $ a_0 (omega), a_1 (omega), a_2 (omega), dots, a_n (omega)$ be a sequence of independent random variables defined on a fixed probability space $(Omega, Pr, A)$. There are many known results for the expected number of real zeros of a polynomial $ a_0 (omega) psi_0(x)+ a_1 (omega)psi_1 (x)+, a_2 (omega)psi_2 (x)+...

On Crossings of Arbitrary Curves by Certain Gaussian Processes1

under conditions which are very close to the necessary ones. Here N(T) is the number of crossings of the level u in (0, T) by the stationary Gaussian process x(t), with covariance function r(r). The symbol £ denotes expectation. The treatment of this problem given by Bulinskaya is essentially a rigorization of the method used by Grenander and Rosenblatt [4], which in turn extends an argument du...