This chapter develops some classical theory and fundamental tools for Gaussian random processes. We start with the basic definitions of Gaussian processes indexed by abstract parameter spaces and, by way of introduction to the subject, derive some elementary yet powerful properties. We present the isoperimetric and log-Sobolev inequalities for Gaussian measures in ℝn and apply them to establish concentration properties for the supremum of a Gaussian process about its median and mean, which are some of the deepest and most useful results on Gaussian processes. Then we introduce Dudley's metric entropy bounds for moments of suprema of (sub-) Gaussian processes as well as for their a.s. modulus of continuity. The chapter also contains a thorough discussion of convexity and comparison properties of Gaussian measures and of reproducing kernel Hilbert spaces and ends with an exposition of the limit theory for suprema of stationary Gaussian processes.

Definitions, Separability, 0-1 Law, Concentration

We start with some preliminaries about stochastic processes, mainly to fix notation and terminology. Then these concepts are specialised to Gaussian processes, and some first properties of Gaussian processes are developed. The fundamental observation is that a Gaussian process X indexed by a a set T induces an intrinsic distance dX on T (dX(s,t) is the L2-distance between X(s) and X(t)), and all the probabilistic information about X is contained in the metric or pseudo-metric space (T,d). This is tested on some of the first properties, such as the 0-1 law and the existence of separable versions of X. One of the main properties of Gaussian processes, namely, their concentration about the mean, is introduced; this subject will be treated in the next section, but a first result on it, which is not sharp but that has been chosen for its simplicity, is given in this section.