Gaussian Processes and the Generic Chaining

Abstract

Chapter 2 describes the “generic chaining” method, a kind of optimal organization of Kolmogorov’s chaining, which forms the core of the book. In its simplest form it allows to bound a stochastic process under a condition on the tails of its increments. The typical case is that of Gaussian processes. These satisfy the following “increment condition”: for u>0 and any s,t∈T, we have P(|X _s −X _t |≥u)≤2exp(−u ²/2d(s,t)²), where d ²(s,t)=E(X _s −X _t )². This simply expresses a (somewhat suboptimal) bound on the tail of the Gaussian r.v. X _s −X _t . The generic chaining bound involves certain “geometric” characteristics of the metric space (T,d) which in a sense measure the “size” of this space in a more precise manner than the classical Dudley’s integral. A fruitful formulation of the generic chaining bound involves sequences of partitions of the metric space, and we provide the basic tools to construct such partitions. We also give a number of concrete examples by analysing ellipsoids in Hilbert space, and showing that Dudley’s integral is already insufficient to study these.