Weak convergence for infinite sum of random variables

Question

Given two independent sequences of random variables $(X^1_k)_{k=1}^\infty, (X^2_k)_{k=1}^\infty,$ respectively converging to $X^1$, $X^2$ in distribution, we know that $(X^1_k + X^2_k)_{k=1}^\infty$ converges to $X^1+X^2$ in distribution. If we have now countably infinite number of independent sequences of random variables $\{(X^j_{k})_{k=1}^\infty\}_{j=1}^\infty$, where each sequence converges in distribution to $X^j$, can we claim the sequence $S_k = \sum_{j=1}^\infty X^j_k$ converges to $X^1 + X^2 + \cdots$ in distribution?

Answer 1

The answer is no.

Intuitively, you can see that there is no claim regrading the rate of convergence of any of the sequences, thus they can converge "as slow as we want", which will make the sum of all sequences (in the same index) ambiguous. A concrete example can be (of constant random variables, for simplicity)

$$ X_k ^ j = 1_{k\le j} $$

Each of the sequences of course converges to $0$ since it is constant for $k$ large enough, and their sum is $S_k = \sum_{j=1}^\infty X_k^j = k$, which diverge.