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Given independent real random variables $X_1,X_2,... \in L^2$ with $$\sum_{n=0}^\infty \frac{Var[S_n]}{n^2}<\infty$$ (here $S_n := X_1+...+X_n$).

How do you show that $(X_n)_{n \in \Bbb N}$ fulfills the strong law of large numbers?

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See Chapter X of Feller, "Introduction to Probability Theory and its Applications, Vol. I, Wiley (1968). He proves the theorem on pages 259-261.

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