Expected Number of Local Maxima of Some Gaussian Random Polynomials

Abstract

Let $Q_n(x)={\sum }_{i=0}^n{A}_i{x}^i$ be a random algebraic polynomial where the coefficients $A_0,A_1,...$ form a sequence of centered Gaussian random variables. Moreover, assume that the increments ${\Delta }_j=A_j-A_{j-1}$, $j=0,1,2,...$ are independent, $A_{-1}=0$. The coefficients can be considered as $n$ consecutive observations of a Brownian motion. We study the asymptotic behaviour of the expected number of local maxima of $Q_n(x)$ below level $u=O(n^k)$, for some $k>0$.

12 pages