From the standard literature it is well known that for sequences of random variables $X_{1, n} \stackrel{P}{\rightarrow} X_1$ and $X_{2, n} \stackrel{P}{\rightarrow} X_2$ as $n \rightarrow \infty$ it holds that $(X_{1, n}, X_{2, n}) \stackrel{P}{\rightarrow} (X_1, X_2)$ for $n \rightarrow \infty$. Using a continuous mapping theorem argument this can be used to establish that $X_{1, n} + X_{2, n} \stackrel{P}{\rightarrow} X_1 + X_2$ for $n \rightarrow \infty$.

Question in general case

Under which conditions can the arguments of the standard literature be extended from a simple sum of two sequences to a sum of sequences whose number of summands also increases to infinity as $n$ goes to infinity. Specifically, given a sequences $X_{l, n}$, $l = 1, \ldots, n$, with $X_{l, n} \stackrel{P}{\rightarrow} X_l$ for all $l = 1, \ldots, n$ under which conditions does it holds that $$ \lim_{n \rightarrow \infty} \sum_{l = 1}^n X_{l, n} \stackrel{P}{\rightarrow} \sum_{l = 1}^\infty X_{l} $$ as $n \rightarrow \infty$?

Question in special case

In my particular use case something more specific would also suffice. Though, I think the more general question is also interesting. If we know that $X_{l, n}$, $l = 1, \ldots, n$, are nonnegative random sequences, i.e. each sequence element is a nonnegative random variable, and $X_{l, n} \stackrel{P}{\rightarrow} 0$ for all $l = 1, \ldots, n$ under which conditions does it holds that $$ \lim_{n \rightarrow \infty} \sum_{l = 1}^n X_{l, n} \stackrel{P}{\rightarrow} 0 $$ as $n \rightarrow \infty$?