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 $k>0$.
Categories: math.PR