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

We study a random algebraic polynomial Qn(x)=∑i=0nAixi, where the coefficients A0,A1,… form a sequence of centred Gaussian random variables. Moreover, we assume that the increments Δj=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 behaviour of the expected number of times that such a random polynomial assumes the real value K, where K is any non-zero real constant. It is shown that the results are valid even for K→∞, as long as K=o(n1/4).