Dyadic quadratic variation of Brownian motion ($\mathbb{E}(Q_n-1{)}^2=2^{1-n}$)

On $[0,1]$, the sum $Q_n$ of squared adjacent dyadic increments of standard Brownian motion has $\mathbb{E}{Q}_n=1$ and $\mathrm{var}(Q_n)=2^{1-n}$. Therefore $Q_n\to 1$ 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.