EXPECTED NUMBER OF LOCAL MAXIMA OF SOME GAUSSIAN RANDOM POLYNOMIALS

Let Q_n(x) =Sⁿ_i=0  A_ixⁱ be a random algebraic polynomial where the coefficients A₀,A₁, · · · form a sequence of centered Gaussian random variables. Moreover, assume that the increments Dj = A_j - A_j−1, j = 0, 1, 2, · · · , with A₋₁ = 0, are independent. 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.