I've seen the law of large numbers stated mainly in two (or three) forms: $S_n/n$ converges in probability (weak law) and converges almost surely (strong law). Also, there is convergence in the $L^2$-norm for uncorrelated random variables ($L^2$ weak law).
However there is a backwards martingale proof of the strong law of large numbers (any graduate level probability theory book, for example Durrett, should have it). The important thing is that $M_{-n}:=S_n/n$ is a backward martingale, and backward martingales converge both a.e. and in the $L^1$-norm. Then, in particular, $S_n/n$ converges in the $L^1$-norm.
Does $S_n/n$ really converge in the $L^1$-norm?
- If yes, why is this never mentioned?
- If no, what is wrong with my above proof?