  We prove a central limit theorem for the mesoscopic linear statistics of
$N\times N$ Wigner matrices $H$ satisfying $\mathbb{E}|H_{ij}|^2=1/N$ and
$\mathbb{E} H_{ij}^2= \sigma /N$, where $\sigma \in [-1,1]$. We show that on
all mesoscopic scales $\eta$ ($1/N \ll \eta \ll 1$), the linear statistics of
$H$ have a sharp transition at $1-\sigma \sim \eta$. As an application, we
identify the mesoscopic linear statistics of Dyson's Brownian motion $H_t$
started from a real symmetric Wigner matrix $H_0$ at any nonnegative time $t
\in [0,\infty]$. In particular, we obtain the transition from the central limit
theorem for GOE to the one for GUE at time $t \sim \eta$.
