For a fixed $t\in [0,1]$ I have a sequence $(X^t_n)_{n\geq 1}$ of normal distributed random variables which is a martingale and bounded in $L^2$. So by the martingale convergence theorem there exists a r.v. $X^t\in L^2$ such that $X^t_n\rightarrow X^t$ $\mathbb{P}$-a.s. and in $L^2$. For this reason we have also that the sequence is uniformly integrable. Hence dominated convergence apply
Consider now the process $X_n^tX_n^s$ for another fixed $s\in [0,1]$. I would like to apply on this product the dominated convergence also. But I have a problem by dominating the sequence. Uniform integrability I think goes here lost. How can I proceed to use DCT?
One can show the convergence in $\mathbb L^1$ of $(X_n^tX_n^s)$ to $X^tX^s$ by the estimates $$\mathbb E|X_n^tX_n^s -X^tX^s|\leqslant \mathbb E|(X_n^t-X^t)X_n^s |+\mathbb E|X^t(X_n^s -X^s)|\leqslant \lVert X_n^s\rVert_2\lVert X_n^t-X^t\rVert_2+\lVert X^t\rVert_2\lVert X_n^s-X^s\rVert_2$$ and boundedness of the sequence $\left( \lVert X_n^s\rVert_2\right)_{n\geqslant 1}$.