Bounds for expected maxima of Gaussian processes and their discrete approximations

Abstract

The paper deals with the expected maxima of continuous Gaussian processes X = (Xt)t≥0 that are Hölder continuous in L2-norm and/or satisfy the opposite inequality for the L2-norms of their increments. Examples of such processes include the fractional Brownian motion and some of its “relatives” (of which several examples are given in the paper). We establish upper and lower bounds for Emax≤t≤1 Xt and investigate the rate of convergence to that quantity of its discrete approximation Emax0≤i≤n Xi/n. Some further properties of these two maxima are established in the special case of the fractional Brownian motion.