On the rate of convergence to the semi-circular law

Abstract

Let $\mathbf{X}=(X_{jk})$ denote a Hermitian random matrix with entries $X_{jk}$, which are independent for $1\le j\le k$. We consider the rate of convergence of the empirical spectral distribution function of the matrix $\mathbf{X}$ to the semi-circular law assuming that $\mathbf{E}{X}_{jk}=0$, $\mathbf{E}{X}_{jk}^2=1$ and that the distributions of the matrix elements $X_{jk}$ have a uniform sub exponential decay in the sense that there exists a constant $\varkappa >0$ such that for any $1\le j\le k\le n$ and any $t\ge 1$ we have $$\mathrm{Pr}\{|X_{jk}|>t\}\le {\varkappa }^{-1}\exp \{-t^{\varkappa }\}.$$ By means of a recursion argument it is shown that the Kolmogorov distance between the empirical spectral distribution of the Wigner matrix $\mathbf{W}=\frac{1}{\sqrt{n}}\mathbf{X}$ and the semicircular law is of order $O(n^{-1}{\log }^{b}n)$ with some positive constant $b>0$.

arXiv commentThis version fills a gap in the previous version using martingale large deviation bounds. Here iterative expansions are used in Section 4, Proposition 4.1 with supporting bounds in Section 3, Lemmas 3.2--3.9, together with expansion of resolvents in rows and columns and re-expressing some terms again in the original resolvents (e.g. (3.45)) and Lemmas 3.10--3.19