In this post, we present two important results of the field of Gaussian Processes: the Slepian and Gordon Theorems. For both of them, we illustrate their power giving specific applications to random matrix theory. More precisely, we derive bounds on the expected operator norm of gaussian matrices.

## 1. Slepian’s Theorem and application

**Theorem (Slepian Fernique 71′)**

Let and be two ** centered **gaussian vectors such that

Then,

**Theorem (Slepian 65′)**

We keep assumptions and notations of the Slepian-Fernique Theorem and we ask further that

Then it holds

**Remarks:**

The assumptions of the Slepian’s Theorem, namely and are equivalent to the following condition

Stated otherwise, the covariance matrices and of and are entrywise comparable with the coefficient of greater or equal to those of with the same variances.

Note that since

the conclusion of the Slepian’s Theorem is stronger compared to the one of the Slepian-Fernique’s Theorem.

**Consequence**

Let us consider a random variable variables taking values in with entries . Then, denoting the operator norm, it holds

where for all and for all .

Moreover,

i.e.

*Proof.*

Let us recall that

Hence it holds

where for all ,

Let us first consider finite sets and . Then we get for any and any ,

where in the last equality, we used the fact that and have a -norm equal to . Hence we proved that ,

Now, let us introduce for any the random variable

with . Then,

leading to

We can then apply the Slepian Fernique theorem to obtain that

To conclude the proof, we need to show that the previous computations derived with finite sets and are enough to get a bound on the operator norm (for which the supremum run on both and ). For this, we need to introduce the notion of covering sets.

**Definition**

Let us consider and . A set is called an -net (covering) of , if

Then, the following Lemma proves that it is sufficient to work with supremum on finite sets rather than the whole continuous spaces and . By taking the limit , the Lemma and the previous computations directly give the stated bound for the expected operator norm of a gaussian matrix .

**Lemma**

There exists an -net for of size

For an -net,

## 2. Gordon’s Theorem and application

**Gordon’s Theorem**Let and be two

*gaussian processes such that*

**centered**

Then

**Remarks**

If we apply Gordon’s inequality for and , we get . Hence Gordon’s inequality contains Slepian’s inequality by taking the second index set to be a singleton set.

**Consequence**

For a given matrix , if (meaning that the linear map is injective), then the map

is an isomorphism with inverse denoted . The operator norm of is given by

Hence, the previous expression allows us to understand that the Gordon Theorem can be a convenient tool to upper bound the expected value of the inverse of the operator norm of the inverse of a gaussian matrix . Following an approach similar to the previous section, one can prove that

with

**Remark**

Let us finally point out that the approaches used in the proofs can allow to get a uniform bound for a random quadratic form given by

where and are arbitrary bounded sets. Such inequality can be given by the

**Chevet’s inequality**which states that given a matrix whose entries are independent, mean zero, sub-gaussian random variables, it holds

where is an absolute constant and where is the maximum of the 2-Orlicz norms of the entries of . and are respectively the gaussian width and the radius of the set and are defined as follows

with . and are defined analogously.

We refer to Section 8.7 of the book* High-Dimensional Probability* from R.Vershynin for details.