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Consider the sum ${S_n := X_1+\ldots+X_n}$ of iid real random variables ${X_1,\ldots,X_n \equiv X}$ of finite mean ${\mu}$ and variance ${\sigma^2}$ for some ${\sigma > 0}$. Then the sum ${S_n}$ has mean ${n\mu}$ and variance ${n\sigma^2}$, and so (by Chebyshev’s inequality) we expect ${S_n}$ to usually have size ${n\mu + O(\sqrt{n} \sigma)}$. To put it another way, if we consider the normalised sum

$\displaystyle Z_n := \frac{S_n - n \mu}{\sqrt{n} \sigma} \ \ \ \ \ (1)$

then ${Z_n}$ has been normalised to have mean zero and variance ${1}$, and is thus usually of size ${O(1)}$.

In the previous set of notes, we were able to establish various tail bounds on ${Z_n}$. For instance, from Chebyshev’s inequality one has

$\displaystyle {\bf P}(|Z_n| > \lambda) \leq \lambda^{-2}, \ \ \ \ \ (2)$

and if the original distribution ${X}$ was bounded or subgaussian, we had the much stronger Chernoff bound

$\displaystyle {\bf P}(|Z_n| > \lambda) \leq C \exp( - c \lambda^2 ) \ \ \ \ \ (3)$

for some absolute constants ${C, c > 0}$; in other words, the ${Z_n}$ are uniformly subgaussian.

Now we look at the distribution of ${Z_n}$. The fundamental central limit theorem tells us the asymptotic behaviour of this distribution:

Theorem 1 (Central limit theorem) Let ${X_1,\ldots,X_n \equiv X}$ be iid real random variables of finite mean ${\mu}$ and variance ${\sigma^2}$ for some ${\sigma > 0}$, and let ${Z_n}$ be the normalised sum (1). Then as ${n \rightarrow \infty}$, ${Z_n}$ converges in distribution to the standard normal distribution ${N(0,1)_{\bf R}}$.

Exercise 1 Show that ${Z_n}$ does not converge in probability or in the almost sure sense (in the latter case, we think of ${X_1,X_2,\ldots}$ as an infinite sequence of iid random variables). (Hint: the intuition here is that for two very different values ${n_1 \ll n_2}$ of ${n}$, the quantities ${Z_{n_1}}$ and ${Z_{n_2}}$ are almost independent of each other, since the bulk of the sum ${S_{n_2}}$ is determined by those ${X_n}$ with ${n > n_1}$. Now make this intuition precise.)

Exercise 2 Use Stirling’s formula from Notes 0a to verify the central limit theorem in the case when ${X}$ is a Bernoulli distribution, taking the values ${0}$ and ${1}$ only. (This is a variant of Exercise 2 from those notes, or Exercise 2 from Notes 1. It is easy to see that once one does this, one can rescale and handle any other two-valued distribution also.)

Exercise 3 Use Exercise 9 from Notes 1 to verify the central limit theorem in the case when ${X}$ is gaussian.

Note we are only discussing the case of real iid random variables. The case of complex random variables (or more generally, vector-valued random variables) is a little bit more complicated, and will be discussed later in this post.

The central limit theorem (and its variants, which we discuss below) are extremely useful tools in random matrix theory, in particular through the control they give on random walks (which arise naturally from linear functionals of random matrices). But the central limit theorem can also be viewed as a “commutative” analogue of various spectral results in random matrix theory (in particular, we shall see in later lectures that the Wigner semicircle law can be viewed in some sense as a “noncommutative” or “free” version of the central limit theorem). Because of this, the techniques used to prove the central limit theorem can often be adapted to be useful in random matrix theory. Because of this, we shall use these notes to dwell on several different proofs of the central limit theorem, as this provides a convenient way to showcase some of the basic methods that we will encounter again (in a more sophisticated form) when dealing with random matrices.

Van Vu and I have just uploaded to the arXiv our paper “Random matrices: universality of local eigenvalue statistics“, submitted to Acta Math..  This paper concerns the eigenvalues $\lambda_1(M_n) \leq \ldots \leq \lambda_n(M_n)$ of a Wigner matrix $M_n = (\zeta_{ij})_{1 \leq i,j \leq n}$, which we define to be a random Hermitian $n \times n$ matrix whose upper-triangular entries $\zeta_{ij}, 1 \leq i \leq j \leq n$ are independent (and whose strictly upper-triangular entries $\zeta_{ij}, 1 \leq i < j \leq n$ are also identically distributed).  [The lower-triangular entries are of course determined from the upper-triangular ones by the Hermitian property.]  We normalise the matrices so that all the entries have mean zero and variance 1.  Basic examples of Wigner Hermitian matrices include

1. The Gaussian Unitary Ensemble (GUE), in which the upper-triangular entries $\zeta_{ij}, i are complex gaussian, and the diagonal entries $\zeta_{ii}$ are real gaussians;
2. The Gaussian Orthogonal Ensemble (GOE), in which all entries are real gaussian;
3. The Bernoulli Ensemble, in which all entries take values $\pm 1$ (with equal probability of each).

We will make a further distinction into Wigner real symmetric matrices (which are Wigner matrices with real coefficients, such as GOE and the Bernoulli ensemble) and Wigner Hermitian matrices (which are Wigner matrices whose upper-triangular coefficients have real and imaginary parts iid, such as GUE).

The GUE and GOE ensembles have a rich algebraic structure (for instance, the GUE distribution is invariant under conjugation by unitary matrices, while the GOE distribution is similarly invariant under conjugation by orthogonal matrices, hence the terminology), and as a consequence their eigenvalue distribution can be computed explicitly.  For instance, the joint distribution of the eigenvalues $\lambda_1(M_n),\ldots,\lambda_n(M_n)$ for GUE is given by the explicit formula

$\displaystyle C_n \prod_{1 \leq i (0)

for some explicitly computable constant $C_n$ on the orthant $\{ \lambda_1 \leq \ldots \leq \lambda_n\}$ (a result first established by Ginibre).  (A similar formula exists for GOE, but for simplicity we will just discuss GUE here.)  Using this explicit formula one can compute a wide variety of asymptotic eigenvalue statistics.  For instance, the (bulk) empirical spectral distribution (ESD) measure $\frac{1}{n} \sum_{i=1}^n \delta_{\lambda_i(M_n)/\sqrt{n}}$ for GUE (and indeed for all Wigner matrices, see below) is known to converge (in the vague sense) to the Wigner semicircular law

$\displaystyle \frac{1}{2\pi} (4-x^2)_+^{1/2}\ dx =: \rho_{sc}(x)\ dx$ (1)

as $n \to \infty$.  Actually, more precise statements are known for GUE; for instance, for $1 \leq i \leq n$, the $i^{th}$ eigenvalue $\lambda_i(M_n)$ is known to equal

$\displaystyle \lambda_i(M_n) = \sqrt{n} t(\frac{i}{n}) + O( \frac{\log n}{n} )$ (2)

with probability $1-o(1)$, where $t(a) \in [-2,2]$ is the inverse cumulative distribution function of the semicircular law, thus

$\displaystyle a = \int_{-2}^{t(a)} \rho_{sc}(x)\ dx$.

Furthermore, the distribution of the normalised eigenvalue spacing $\sqrt{n} \rho_{sc}(\frac{i}{n}) (\lambda_{i+1}(M_n) - \lambda_i(M_n))$ is known; in the bulk region $\varepsilon n \leq i \leq 1-\varepsilon n$ for fixed $\varepsilon > 0$, it converges as $n \to \infty$ to the Gaudin distribution, which can be described explicitly in terms of determinants of the Dyson sine kernel $K(x,y) := \frac{\sin \pi(x-y)}{\pi(x-y)}$.  Many further local statistics of the eigenvalues of GUE are in fact governed by this sine kernel, a result usually proven using the asymptotics of orthogonal polynomials (and specifically, the Hermite polynomials).  (At the edge of the spectrum, say $i = n-O(1)$, the asymptotic distribution is a bit different, being governed instead by the  Tracy-Widom law.)

It has been widely believed that these GUE facts enjoy a universality property, in the sense that they should also hold for wide classes of other matrix models. In particular, Wigner matrices should enjoy the same bulk distribution (1), the same asymptotic law (2) for individual eigenvalues, and the same sine kernel statistics as GUE. (The statistics for Wigner symmetric matrices are slightly different, and should obey GOE statistics rather than GUE ones.)

There has been a fair body of evidence to support this belief.  The bulk distribution (1) is in fact valid for all Wigner matrices (a result of Pastur, building on the original work of Wigner of course).  The Tracy-Widom statistics on the edge were established for all Wigner Hermitian matrices (assuming that the coefficients had a distribution which was symmetric and decayed exponentially) by Soshnikov (with some further refinements by Soshnikov and Peche).  Soshnikov’s arguments were based on an advanced version of the moment method.

The sine kernel statistics were established by Johansson for Wigner Hermitian matrices which were gaussian divisible, which means that they could be expressed as a non-trivial linear combination of another Wigner Hermitian matrix and an independent GUE.  (Basically, this means that distribution of the coefficients is a convolution of some other distribution with a gaussian.  There were some additional technical decay conditions in Johansson’s work which were removed in subsequent work of Ben Arous and Peche.)   Johansson’s work was based on an explicit formula for the joint distribution for gauss divisible matrices that generalises (0) (but is significantly more complicated).

Just last week, Erdos, Ramirez, Schlein, and Yau established sine kernel statistics for Wigner Hermitian matrices with exponential decay and a high degree of smoothness (roughly speaking, they require  control of up to six derivatives of the Radon-Nikodym derivative of the distribution).  Their method is based on an analysis of the dynamics of the eigenvalues under a smooth transition from a general Wigner Hermitian matrix to GUE, essentially a matrix version of the Ornstein-Uhlenbeck process, whose eigenvalue dynamics are governed by Dyson Brownian motion.

In my paper with Van, we establish similar results to that of Erdos et al. under slightly different hypotheses, and by a somewhat different method.  Informally, our main result is as follows:

Theorem 1. (Informal version)  Suppose $M_n$ is a Wigner Hermitian matrix whose coefficients have an exponentially decaying distribution, and whose real and imaginary parts are supported on at least three points (basically, this excludes Bernoulli-type distributions only) and have vanishing third moment (which is for instance the case for symmetric distributions).  Then one has the local statistics (2) (but with an error term of $O(n^{-1+\delta})$ for any $\delta>0$ rather than $O(\log n/n)$) and the sine kernel statistics for individual eigenvalue spacings $\sqrt{n} \rho_{sc}(\frac{i}{n}) (\lambda_{i+1}(M_n) - \lambda_i(M_n))$ (as well as higher order correlations) in the bulk.

If one removes the vanishing third moment hypothesis, one still has the sine kernel statistics provided one averages over all i.

There are analogous results for Wigner real symmetric matrices (see paper for details).  There are also some related results, such as a universal distribution for the least singular value of matrices of the form in Theorem 1, and a crude asymptotic for the determinant (in particular, $\log |\det M_n| = (1+o(1)) \log \sqrt{n!}$ with probability $1-o(1)$).

The arguments are based primarily on the Lindeberg replacement strategy, which Van and I also used to obtain universality for the circular law for iid matrices, and for the least singular value for iid matrices, but also rely on other tools, such as some recent arguments of Erdos, Schlein, and Yau, as well as a very useful concentration inequality of Talagrand which lets us tackle both discrete and continuous matrix ensembles in a unified manner.  (I plan to talk about Talagrand’s inequality in my next blog post.)