Brownian weighted quadratic variation in L2 (${\sum }_j{B}_{t_j}(B_{t_{j+1}}-B_{t_j}{)}^2⟶{\int }_0^t{B}_{s}ds$)

For Brownian motion and deterministic partitions of an interval $[0,t]$ whose mesh $|\pi |$ tends to zero, the displayed limit holds in L2 space. Split the error into $U_{\pi }={\sum }_j{B}_{t_j}((\Delta B_j{)}^2-\Delta t_j)$ and the Riemann sum error $V_{\pi }={\sum }_j{B}_{t_j}\Delta t_j-{\int }_0^t{B}_{s}ds$. The summands of $U_{\pi }$ are martingale differences, hence orthogonal in L2 space, and the Gaussian fourth moment givesThe Cauchy-Schwarz inequality and the Brownian increment variance give $\mathbb{E}{V}_{\pi }^2\le (t/2){\sum }_j(\Delta t_j{)}^2\le t^2|\pi |/2$. Both vanish. This proves the limit directly without a stochastic change-of-variables formula.