  Let $\mathbf X=(X_{jk})_{j,k=1}^n$ denote a Hermitian random matrix with
entries $X_{jk}$, which are independent for $1\le j\le k\le n$. 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 $$ \sup_{n\ge1}\sup_{1\le j,k\le
n}{\mathbf E}|X_{jk}|^4=:\mu_4<\infty \quad \text{and} \sup_{1\le j,k\le
n}|X_{jk}|\le D_0n^{\frac14}. $$ By means of a recursion argument it is shown
that the Kolmogorov distance between the expected spectral distribution of the
Wigner matrix $\mathbf W=\frac1{\sqrt n}\mathbf X$ and the semicircular law is
of order $O(n^{-1})$.
