Expected Number of Slope Crossings of Certain Gaussian Random Polynomials

Abstract

Let Q n (x) = n i=0 A i x i be a random polynomial where the coefficients A 0 , A 1 , · · · form a sequence of centered Gaussian random variables. Moreover, assume that the increments ∆ j = A j − A j−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 crosses a line which is not necessarily parallel to the x-axis. More precisely we obtain the asymptotic behavior of the expected number of real roots of the equation Q n (x) = Kx, for the cases that K is any non-zero real constant K = o(n 1/4 ), and K = o(n 1/2 ) separately.

Average Number of Real Zeros of Random Algebraic Polynomials Defined by the Increments of Fractional Brownian Motion

“…One would mention among others the problem of non-zero crossings for random polynomials studied by Dembo and Mukherjee [3], the problem of random polynomials having few or no real zeros (Dembo et al [2]), the problem of distribution of zeros of random analytic functions (Kabluchko and Zaporozhets [8]) and the problem of real zeros of linear combinations of orthogonal polynomials (Lubinsky, Pritsker and Xie [12]). It would also be interesting to study possible extension the results of Rezakhah and Shemehsavar [16,17] on Brownian motion to the general case of fractional Brownian motion.…”

“…Rezakhah and Shemehsavar [16,17] studied random polynomials Q n (x) = n−1 i=0 A i x i where the coefficients A i are successive Brownian images A i = W (t i ), t 0 < t 1 < . .…”

New features on real zeros of random polynomials

“…Recently there has been much interest in cases where the coefficients form certain random processes, see for e.g. Rezakhah and Soltani [11,12]; Rezakhah and Shemehsavar [13,14].…”