  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 $$ \Pr\{|X_{jk}|>t\}\le \varkappa^{-1}\exp\{-t^{\varkappa}\}. $$ By means
of a recursion argument it is shown that the Kolmogorov distance between the
empirical spectral distribution of the Wigner matrix $\mathbf W=\frac1{\sqrt
n}\mathbf X$ and the semicircular law is of order $O(n^{-1}\log^b n)$ with some
positive constant $b>0$.
