Expected Number of Slope Crossings of Certain Gaussian Random Polynomials

Let Q(n)(x) = Sigma(n)(i=0) = A(i)x(i) be a random 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 study the number of times that such a random polynomial crosses a line which is not necessarily parallel to the x-axis. More precisely we obtain the asymptotic behavior of the expected number of real roots of the equation Q(n)(x) = Kx, for the cases that K is any non-zero real constant K = o(n(1/4)), and K = o(n(1/2)) separately.