# Simulation of the semicircle Wigner Theorem

Let's begin with some termes:

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.

1. 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.

1. 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.

• What is your question? – rm -rf Mar 9 '15 at 4:09
• In the linked question, you already seem to have a simulation. So what is the Mathematica issue you want to address here? – Jens Mar 9 '15 at 5:14
• My question is to do the same thing that was done for the central limit theorem as it is shown in this page demonstrations.wolfram.com/TheCentralLimitTheorem – Zbigniew Mar 9 '15 at 6:51