From the standard literature it is well known that for sequences of random variables X1,nP→X1 and X2,nP→X2 as n→∞ it holds that (X1,n,X2,n)P→(X1,X2) for n→∞. Using a continuous mapping theorem argument this can be used to establish that X1,n+X2,nP→X1+X2 for n→∞.
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 Xl,n, l=1,…,nl∈N, with Xl,nP→Xl (n→∞) for all l=1,…,nl∈N under which conditions does it holds that limn→∞n∑l=1Xl,nP→∞∑l=1Xl as n→∞?
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 Xl,n, l=1,…,nl∈N, are nonnegative random sequences, i.e. each sequence element is a nonnegative random variable, and Xl,nP→0 (n→∞) for all l=1,…,nl∈N under which conditions does it holds that limn→∞n∑l=1Xl,nP→0 as n→∞? Possibly, what would also be interesting and might be easier to answer is when the sample mean converges to zero, i.e., limn→∞1nn∑l=1Xl,nP→0 as n→∞