Some references
- "Berry-Esseen bounds and almost sure CLT for the quadratic variation of a general Gaussian process"
- "Error bounds on the non-normal approximation of Hermite power variations of fractional Brownian motion"
- "Berry–Esséen bounds and almost sure CLT for the quadratic variation of the sub-fractional Brownian motion"
They study this rate of convergence for fractional-BM. For the particular case of standard Brownian motion, we have
$$\frac{V^{n}_{t}-t}{\sigma\sqrt{n}}\sim N(0,1),$$
where $V^{n}_{t}=\sum^{n}_{k=0} (B_{k+1}-B_{k})^{2}$.