On the Average Number of Local Maxima of Certain Gaussian Random Polynomials
by
Saeid Rezakhah
AmirKabir University of Technology - Tehran- Iran

We study a random algebraic polynomial Q_n(x)=∑_i=0ⁿ A_ixⁱ where the coefficients A₀, A₁, ... form a sequence of centered Gaussian random variables. Moreover, assume that the increments D_j=A_j-A_j-1, j=0, 1, 2, ... are independent, A_-1=0. The coefficients can be considered as n consecutive observations of a Brownian motion. We obtain the asymptotic behaviour of the expected number of local maxima of polynomial Q_n(x) below a level u=O(n^k), for some k > 0.

Date received: March 18, 2007