I am familiar with computing the quadratic variation of Brownian motion, but was confused when the text I'm working through introduced cross variation of independent Brownian motions. the notation is as follows:
$$\langle X,Y\rangle_t = \lim_{||\Delta||\to 0} \sum_i(X_{t_{i+1}}-X_{t_i})(Y_{t_{i+1}}-Y_{t_i}) $$
Where $X_t$ and $Y_t$ are independent Brownian motions and $\Delta$ is a partition of $[0,t]$. I believe to proceed I should try to calculate the $L^2$ limit (as hinted at in the text), but I am not sure where to start here. The issue is that the only way I know to prove that $X_n\to X$ in $L^2$ is by showing that $E[| X_n-X|^2]\to0$, but I don't know what to use for $X$ here since I'm trying to compute the limit. Any help is appreciated.