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.
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Talagrand, M. (2014). Gaussian Processes and the Generic Chaining. In: Upper and Lower Bounds for Stochastic Processes. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics, vol 60. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-54075-2_2
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