On the rate of convergence to the semi-circular law

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 $\mathrm{\varkappa }>0$ such that for any $1\le j\le k\le n$ and any $t\ge 1$ we have $$\mathrm{Pr}\{\mathrm{\mid }{X}_{jk}\mathrm{\mid }>t\}\le {\mathrm{\varkappa }}^{-1}\exp \{-t^{\mathrm{\varkappa }}\}\mathrm{.}$$ By means of a recursion argument it is shown that the Kolmogorov distance between the empirical spectral distribution of the Wigner matrix $\mathbf W=\frac1{\sqrt n}\mathbf X$andthesemicircularlawisoforder$O(n^{-1}\log^b n)$ with some positive constant $b>0$.