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 be a finite metric space. Let
be a Gaussian process where each
is a zero-mean Gaussian random variable. The distance between two points
is the square-root of the covariance between
and
. In this post, we are interested in upper-bounding
.
Question: How large can the quantity be?
In this post we are going to prove the following fact:
where is the distance between the point
from the set
, and
is a specific sequence of sets with
. Constructions of these sets will be discussed in a subsequent post.