On the average number of sharp crossings of certain Gaussian random polynomials

Let Q_n(x)=sum _{k=0}^{n} A_{k}x^{k} be a random algebraic polynomial where the coefficients A0,A1, · · · form a sequence of centered Gaussian random variables. Moreover, assume that the increments
j = Aj ?Aj?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 behavior of the expected number of u-sharp crossings, u > 0, of polynomial Qn(x). We refer to u-sharp crossings as those zero upcrossings with slope greater than u, or those down-crossings with slope smaller than ?u. We consider the cases where u is unbounded and increasing with n, say u = o(n5/4), and u = o(n3/2).