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Thomas Kojar
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Some references

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 $V^{n}_{t}=\sum^{n}_{k=0} (B_{k+1}-B_{k})^{2}$.

Thomas Kojar
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