Optimal Bounds for Convergence of Expected Spectral Distributions to the Semi-Circular Law

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 ${\mathbf E} X_{jk}=0$,${\mathbf E} X_{jk}^2=1$andthat$$ \sup_{n\ge1}\sup_{1\le j,k\le n}{\mathbf E}|X_{jk}|^4=:\mu_4<\infty \quad $\text{and}$ \sup_{1\le j,k\le n}|X_{jk}|\le D_0n^{\frac14}. $$ By means of a recursion argument it is shown that the Kolmogorov distance between the expected spectral distribution of the Wigner matrix $\mathbf{W}=\frac{1}{\sqrt{n}}\mathbf{X}$ and the semicircular law is of order $O(n^{-1})$.