# 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.