Convergence to the semicircle law (Q1105270)

Let \(X_{ij}\) (i,j\(\geq 1)\) be a sequence of i.i.d. random variables with variance 1 and finite fourth moment. For p, \(n=n(p)\geq 1\), denote the \(p\times n\) matrix \((X_{ij})_{1\leq i\leq p, 1\leq j\leq n}\) by \(X_ p\). The authors prove that for \(x\in {\mathbb{R}}\), \[ p^{-1}\#\{p: (1/2\sqrt{np})(X_ p X'_ p-nI_ p)\leq x\} \] converges a.s. to w(x) as \(p\to \infty\), n(p)\(\to \infty\) and p/n(p)\(\to 0\), where w(x) denotes the distribution function of the semicircle law, i.e. has density \(2\pi^{-1}\sqrt{1-x^ 2} 1_{[-1,1]}(x)\).