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

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, 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 number of u-sharp crossings of polynomial $Q_n(x)$ . We refer to u-sharp crossings as those zero up-crossings with slope greater than $u$, or those down-crossings with slope smaller than $-u$. We consider the cases where $u$ is unbounded and is increasing with $n$, where $u=o(n^{5/4})$, and $u=o(n^{3/2})$ separately.