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koobstrukcja

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Series $S_n:=\sum\limits_{k=1}^n X_k$ of independenet random variables converges in distribution. Prove that $S_n$ converges almost certainly.

I suppose these are not sufficent assuptions about $X_n$ and a distriubiution limit $S$ to provide a convergernt martingale related to this sequence. They should at least be integrable...

I've thought about following martingale: $S_n-\mathbb{E}S_n-S+\mathbb{E}S$ adapted to filtration $\mathcal{F}_n=\sigma (S,X_1,X_2,\ldots ,X_n)$, where $S$ denotes a random variable, having distribution $S$, independant from $\sigma (X_1,X_2,\ldots ,X_n\ldots)$, but I can't provide proof of its boundeness in $L^1(\Omega)$.

I know how to prove it without any martigale theorems, in full generality- the first step is proving that the series converge in probability, which might be usefull for a further martingale proof.

Thanks in andvance for any suggestions and ideas.