Is it true that if a sequence of random matrices $\{X_n\}$ converge in probability to a random matrix $X_n\overset{P}{\to}X$ as $n\to\infty$ that the elements $X_n^{(i,j)}\overset{P}{\to} X^{(i,j)}$ $\forall i,j$ also, or are there additional conditions required?
I think I have proved this using the norm $\|A\|=\max_j \sum_i |A^{(i,j)}|$ and the equivalence of norms, however I have only found it stated elsewhere with the caveat that $\{X_n\}$ are symmetric or symmetric and nonnegative-definite.
Yes: if $Y=\max\limits_{1\leqslant k\leqslant N}|Y_k|$ converges to $0$ in probability, then each $Y_k$ converges to $0$ in probability.
Proof: For every $\varepsilon\gt0$, $[Y_k\geqslant\varepsilon]\subseteq[Y\geqslant\varepsilon]$. QED