Bounding the Supremum of a Gaussian Process: Talagrand’s Generic Chaining (Part 1)

This post is part of a series which answers a certain question about the supremum of a Gaussian process. I am going to write, as I have understood, a proof given in Chapter 1 of the book “Generic Chaining” by Michel Talagrand. I recommend the reader to take a look at the excellent posts by James Lee on this matter. (I am a beginner, James Lee is a master.)

Let (T,d) be a finite metric space. Let \{X_t\} be a Gaussian process where each X_t is a zero-mean Gaussian random variable. The distance between two points s,t\in T is the square-root of the covariance between X_s and X_t. In this post, we are interested in upper-bounding Q.

Question: How large can the quantity Q := \mathbb{E} \sup_{t\in T} X_t be?

In this post we are going to prove the following fact:

\boxed{\displaystyle \mathbb{E}\sup_{t\in T}X_t \leq O(1)\cdot  \sup_{t\in T}\sum_{n\geq 1}{2^{n/2}d(t,T_{n-1})} ,}

where (t, A) is the distance between the point X_t from the set A, and \{T_i\} is a specific sequence of sets with T_i\subset T. Constructions of these sets will be discussed in a subsequent post.

Continue reading “Bounding the Supremum of a Gaussian Process: Talagrand’s Generic Chaining (Part 1)”

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