Let $(B_{t_i})_{t_i \in [0, t]}$ be a Brownian Motion process and $Q_n(t) := \sum^{n}_{i=1} (\Delta_i B) ^2$ where $\Delta_i B = B_{t_i} - B_{t_{i-1}}$.
How is it that $Q_n(t) := \sum^{n}_{i=1} (\Delta_i B) ^2$ converges to $t$ in mean-square but for a given sample path $\omega \in \Omega$, we have that $Q_n(t)$ does not converge to any number?
What is actually happening to $Q_n(t)$ as $n \rightarrow \infty$? Is it oscillating around $t$?