Let $X_k, k\geq 1$ be a sequence of random variables and let $S_n:=\sum_{k=1}^n X_k, n \geq 1$ be the sequence of partial sums.

When the $X_k$ are Independent, Kolmogorov's 3-series Theorem gives some result about the P-a.s. convergence of $S_n$.

When the $X_k$ are identically distributed (and some other conditions are met, but in particular they are not independent), then WLLN implies that $S_n/n \rightarrow E[X_k]$ in $L^2$ and in probability. Unless $E[X_k]=0$, this automatically implies that the series $S_n$ diverges? Also, Kolmogorov's 0-1-law implies that in case of Independent $X_k$, we have either P-a.s. convergence or P-a.s divergence. But in that case (when there is no Independence), we can have convergence with probability $p$ and divergence with probability $1-p$, this is no Problem at all?

But if some more conditions are met (the ones for SLLN), then SLLN implies P-a.s. convergence of $S_n/n$ (but not $S_n$). If These conditions are not met, then again convergence with probability $p \in (0,1)$ would be possible?

If we want to know how $S_n$ behaves, then we have to apply CLT?

You see, there is a big mess in my mind about the different methods and I would really appreciate if you could give me an intuitive overview (I know the results, but not the Interpretation) over These methods and when to apply, which Kind of implications there are, ...

Thank you very much!!


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