this is rather general question and there might be no such thing as a right answer, but I would admire any replies.
Consider an iid sequence of random variables $X_1,X_2,\dots$, then Kolmogorov's three series theorem tells us precisely whether or not the partial sum process $S_n = \sum_{i=1}^n X_i$ converges.
However, the most important results regarding the convergence of $S_n$ seem to be the strong law of large numbers and the Central limit theorem.
Why is that? Is it, that $S_n$ does not converge in most cases, but only a rescaled version, for example $\frac{1}{n}S_n$ as in the SLLN. Or is the problem that Kolmogorov's three series theorem not provide any information (in case of convergence) how a limit could look like? Or is it even that for practical purposes the average is more interesting than the partial sum?
Or something completely different? Does anyone have some experiences to share?
Thanks in advance.