Let $\frac{X}{\sqrt n}$ be a Wigner matrix, such that $X_{ij}$ are iid random variables, with mean 0 and variance 1, with $X_{ij} = \overline{X_{ij}}$ for $i > j$. Then we know by Wigner's Theorem that the the spectral measure $L_n$ of $\frac{X}{\sqrt n}$ converges weakly to the semicircle law $f(x) = \mathbb{1}_{[-2,2]}(x)\frac{\sqrt{4 - x^2}}{2 \pi }$.
Can anything be said about the convergence of diagonally shifted and scaled Wigner matrices $$W = \sigma \frac{X}{\sqrt n} + a I_n$$ where $\sigma > 0$ and $a \in \mathbb{R}$?