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
$$\begin{aligned} \sup _{n\ge 1}\sup _{1\le j,k\le n}\mathbf{E}|X_{jk}|^4=:\mu _4<\infty , \end{aligned}$$
and
$$\begin{aligned} \sup _{1\le j,k\le n}|X_{jk}|\le D_0n^{\frac{1}{4}}. \end{aligned}$$
By means of a recursion argument it is shown that the Kolmogorov distance between the expected spectral distribution of the Wigner matrix
\(\mathbf {W}=\frac{1}{\sqrt{n}}\mathbf {X}\) and the semicircular law is of order
\(O(n^{-1})\).