Dyadic quadratic variation of Brownian motion
= Dyadic quadratic variation of Brownian motion
{title2=$\mathbb E(Q_n-1)^2=2^{1-n}$}
On $[0,1]$, the sum $Q_n$ of squared adjacent dyadic increments of standard <Brownian motion> has $\mathbb E Q_n=1$ and $\operatorname{var}(Q_n)=2^{1-n}$. Therefore $Q_n\to1$ in $L^2$. On an interval of length $t$, scaling gives mean $t$ and <variance> $2t^2 2^{-n}$. <Independence> is used within each grid, not between different grids.