This question is posted in mathematica.stackexchange
For
$1\le i<j<\infty$ let
$X_{i,j}$ be independant and indentically
distributed (i.i.d) real random variables with mean
$0$ and variance
$1$ and set
$X_{j,i}=X_{i,j}$. Let
$X_{i,i}$ be i.i.d. real random
variables (with possibly a different distribution) with mean
$0$ and
variance
$1$. Then
$M_n=\left[ X_{i,j}\right]_{i,j=1}^n$ will be a
random
$n\times n$ symmetric matrix.
There are
$n$ random eigenvalues which we will denote by $$
\lambda1\le \lambda2\le \dots \lambdan.
$$ The empirical spectral
measure is $$ \nu^*n=\frac1n \sum{i=1}^n \delta{\lambda_i}. $$
This is a random discrete probability measure which puts
$n^{-1}$
mass to each (random) eigenvalue.
theorem:(Wigner's semicircle law) Let
$$ \nu_n=\frac1n \sum_{i=1}^n \delta_{\frac{\lambda_i}{\sqrt{n}}}. $$ be the
normalized empirical spectral measure. Then as
$n\to \infty$ we have
$$ \nu_n \Rightarrow \nu \qquad \text{a.s.} $$ where
$\nu$ has
density
$$
\frac{d\nu}{dx}=\frac{1}{2\pi} \sqrt{4-x^2}\,\chi_{\{|x|\le 2\}}.
$$ (
$\chi_A$ denote the indicatrice function of
$A$)
Question Using simulation to understand this theorem in other word: Do the same thing that was done for the central limit theorem as it is shown in this page CLT
I have already posted here
about the same subject.
Any help is welcome.