I am trying to solve this exercise, of Shiryaev's book:
Let $\xi_1, \xi_2, \dots$ be a sequence of independent random variables. Show that $\Sigma \xi_n$ converges almost surely iff it converges in probability.
I am having trouble to prove the "if" part. This solution uses the Ottaviani inequality, but we haven't proven this result yet.
My thoughts: there is a subsequence of $S_n=\xi_1+\dots+\xi_n$ that converges almost surely, so there is a subsequence $(S_{n_{k}})$ of $S_n$ that converges almost surely. So we could apply the 3-series theorem, and maybe conclude somehow that the whole sequence must converge. But I am not sure of how to proceed. Any hists?
Thanks.
