I. Wigner’s theorem

**Definition**

Let . The probability distribution on defined by

is called the semi-circular distribution. We denote .

** Wigner’s Theorem**– Let i.i.d. centered, complex valued and with .

– Let i.i.d. centered, real valued with .

– The ‘s and the ‘s () are independent.

– Consider and the hermitian matrices defined by

Then almost surely,

where are the (real) eigenvalues of the matrix .

*Proof*

- Show that semi circular law is characterized by its moments (using a corollary of Carleman theorem). Compute the moments of the smi-circular law (and find the Catalan numbers).
- Compute the moments of the spectral density of the Wigner matrix.
- Show that the moments convergences to the Catalan numbers.
- Convergence of moments + limit distribution characterized by moments -> weak convergence

**Additional results**

- If , then

In particular, - If , then

**Remark**.

We refer to the paper “The eigenvalues of random symmetric matrices” to get an extension of the Wigner’s Theorem.

## II. Marcenko pastur

### 1. Preliminaries

### a) Key results for the proof of the Marcenko-Pastur Theorem

**Definition**

*Let us consider a probability measure on . The Stieltjes transform of denoted is defined by*

**Theorem (Properties of the Stieltjes Transform)**

*Let be some measure on with Stieltjes Transform .*

*is analytic on .*- If Supp() , then .
- .
- .
- For any is bounded and continuous,
- For all points of continuity of ,

**Remark. **The previous Theorem states that the Stieltjes Transform is analytic on and when goes to the real axis, the holomorph property is lost but this allows to recover the measure (see properties 6, 7 and 8).

**Theorem (Weak convergence and pointwise convergence of Stieltjes Transform)***Let us consider probability measures on .*

*If , then**a) Let us consider with an accumulation point. If for all , then*

– there exists a measure satisfying such that

–*b) If it also holds that (i.e. from point 4) of the previous Theorem), then is a probability measure and .*

The previous Theorem can be seen as the counterpart of the famous Levy Theorem. The Levy Theorem states the link between the weak convergence and the pointwise convergence of the characteristic function.

**Definition**

*The characteristic function of a real-valued random variable is defined by*

**Theorem (Levy)***Let us consider real-valued random variables and .*

*If , then**If there exists some function such that**and if is continuous at , then*

– there exists a real valued random variable such that .

– .

Just for the sake of beauty, let me mention the following result that can allow to identify functions that can be written as the Stieltjes Transform of some measure on .

**Theorem (Recognize a Stieltjes Transform)***If satisfies*

*is analytic**There exists such that*

*Then there exists a unique measure on satisfying such that *

* *

*If additionally it holds 4. ,Then .*

**Example**: If is the Stieltjes Transform of some measure then is the Stieltjes Transform of some probability measure on .

### b) Proof of the key preliminary result

We will need the following additional properties in the proof.

**Theorem (Helly’s selection theorem)**

*From every sequence of proba measures , one can extract a subsequence that converges vaguely to a measure . (Note that is not necessary a probability measure).*

**Remark.** vaguely implies for every where

*Proof.*

1) To prove that for any , , we show that for any , and . Let us consider some . Since

and since is continuous and bounded on (because ), the weak convergence of the sequence to ensures that

Using an analogous approach, one can show that

which concludes the proof of the first item of the Theorem.

2a) We consider some set with an accumulation point such that .

By the Helly’s selection Theorem, there exists a subsequence of the sequence of probabilities that converges vaguely to some measure on . Let us consider some . Using the previous remark and since belongs to , we get that

Since is a subsequence of the sequence that converges to , we deduce that

(1)

We know from the properties of the Stieltjes Transform that and are analytic functions on . Hence, (1) and the analytic continuation give

We also get that converges vaguely to . Considering another subsequence that converges vaguely to some measure , the previous arguments prove that

Using again the analytic continuation, we obtain that

leading to . This gives that has a unique accumulation point. A standard argument based on the Helly’s selection Theorem ensures that

2b) This directly follows from the equivalece between points and of the following Lemma.**Lemma**

Let be probability measures and be a measure on . The following statements are equivalent.

.

and .

and is tight, namely for any , there exists some compact set such that

### 2. Marcenko-Pastur Theorem

**Theorem (Marcenko-Pastur)**

*Let us consider a matrix with i.i.d. entries such that*

* *

with and of the same order and the spectral measure of :

Then, almost surely (i.e. for almost every realization),

where is the Marcenko-Pastur distribution

with

In what follows, we denote the Stieltjes Transform of the measure .

with and of the same order and the spectral measure of :

Then, almost surely (i.e. for almost every realization),

where is the Marcenko-Pastur distribution

with

In what follows, we denote the Stieltjes Transform of the measure .

**Remark**

- The behavior of the spectral measure brings information about the vast majority of the eigenvalues but is not affected by some individual eigenvalues’ behavior. For example, one may have without affecting the behaviour of the whole sum.
- The Dirac measure at zero is an artifact due to the dimensions of the matrix if .
- If , that is , then typical from the usual regime “small dimensional data vs large samples”. The support of Marcenko-Pastur distribution concentrates around and

(2)

This will allow to show that there exists some set a countable set with an accumulation point such that almost surely

Indeed, suppose that we know that (2) holds et let us consider some sequence with an accumulation point . One can typical take for all and . Then, (2) ensures that for any , there exists some set with such that

(3)

holds for any . Let us denote now and . We get that and for all , it holds

(note that we used the continuity of the functions and to ensure that (3) implies that for all , it holds

We will then conclude the proof of the Theorem using the Theorem of the previous section (called *Weak convergence and pointwise convergence of Stieltjes Transform*).

To prove (2), we use the decomposition

- To deal with the term , we use the Efron-Stein inequality to show that

which allows to prove that almost surely tends to using Borel-Cantelli Lemma. - We then show that satisfies
(4)

and that satisfies the equation(5)

- To conclude the proof (i.e. to get (2)), we use some “stability” result. More precisely, equations (4) and (5) are close and we need to show that solutions of these equations are as a consequence, close to each other. We formalize this result with the following Lemma.

**Lemma**

Let . We assume that there exists two Stieltjes Transform of probability measures on , denoted , that are respectively the solutions of the equation

where . Then

Let

and

**Theorem (convergence of extremal eigenvalues)**

- If , then
- If , then

**Remark**.

Exactly like with Wigner matrices, when the 4th moment of the random varaibles are not finite, goes to . However, contrary to the Wigner case, for Wishart matrices still converges to a finite value.

**Theorem (Fluctuations of and Tracy-Widom distribution)**We can fully describe the fluctuations of :

where