EXPECTED NUMBER OF LOCAL MAXIMA OF SOME GAUSSIAN RANDOM POLYNOMIALS

Abstract. Let Qn(x) = ^2™_0 Atxt be a random algebraic polynomial where the coefficients A0, Ai, • • • form a sequence of centered Gaussian random variables. Moreover, assume that the increments Aj = Aj — Aj-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 Qn(x) below level u = O(nk), for some k 0.