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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.