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Simulation of Wigner semicircle law

Posted 10 years ago

This question is posted in mathematica.stackexchange

  1. For $1\le i&lt;j<\infty$ 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 $$ \lambda1\le \lambda2\le \dots \lambdan. $$ 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.

  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.

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