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I am trying to prove the following theorem somewhat indicating the relationship between Strong and Weak LLNs:

Let $\{S_n\}$ be the partial sums of a series of independent r.v.'s $\{X_n\}$. Then $S_n/n \rightarrow 0\ a.e.$ if and only if the following 2 conditions are satisfied:

1, $S_n/n\rightarrow 0\quad in\ probability$;

2, $S_{2^n}/2^n\rightarrow 0\quad a.e.$

A possible way maybe to first prove $$ D_n=\max_{2^n\leq k<2^{n+1}}|S_k-S_{2^n}|\rightarrow 0\quad a.e., $$ then we have $$ \frac{|S_k|}{k}\le \frac{|S_{2^n}|+D_n}{2^n}\rightarrow 0\ a.e. \quad \text{for}\quad 2^n\le k<2^{n+1}. $$ By Borel-Cantelli Lemma, it's equivalent to prove $$ \sum_nP\{D_n>2^n\epsilon\}<\infty. $$ But how? Since no information about the moments is given, Chebyshev's inequality seems helpless here.

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