Simulation of Wigner semicircle law

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  1. 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.

  2. There are n random eigenvalues which we will denote by $
    \lambda_1\le \lambda_2\le \dots \lambda_n. $ 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.

  3. 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.