Bounds for expected maxima of Gaussian processes and their discrete approximations

TL;DR

This paper derives bounds for the expected maximum of certain Gaussian processes, like fractional Brownian motion, and analyzes how well discrete maxima approximate the continuous maximum.

Contribution

It provides new upper and lower bounds for expected maxima of H"older continuous Gaussian processes and studies the convergence of their discrete approximations.

Findings

Established bounds for expected maxima of Gaussian processes.

Analyzed convergence rates of discrete maxima to continuous maxima.

Explored properties of maxima in fractional Brownian motion.

Abstract

The paper deals with the expected maxima of continuous Gaussian processes $X = (X_t)_{t\ge 0}$ that are H\"older continuous in $L_2$-norm and/or satisfy the opposite inequality for the $L_2$-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 $E \max_{0\le t\le 1}X_t$ and investigate the rate of convergence to that quantity of its discrete approximation $E \max_{0\le i\le n}X_{i/n}$. Some further properties of these two maxima are established in the special case of the fractional Brownian motion.