In a Post I found it says:
Whenever ${\rm E}(X)$ exists (finite or infinite), the strong law of large numbers holds. That is, if $X_1,X_2,\ldots$ is a sequence of i.i.d. random variables with finite or infinite expectation, letting $S_n = X_1+\cdots + X_n$, it holds $n^{-1}S_n \to {\rm E}(X_1)$ almost surely. The infinite expectation case follows from the finite case by the monotone convergence theorem.
Can someone give a reference/answer to this question?
I want to prove that:
If $EX^{+}_{k}=∞ $ and $EX^{-}_{k}<∞ $ then $n^{−1}S_{n}→∞$ a.s.
Best Answer
The answer is positive. Reference: first page of the paper The Strong Law of Large Numbers When the Mean is Undefined KB Erickson · 1973 TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 185. November 1973
Also theorem 2.4.5, R.Durrett, Probability, theory and examples (there's a proof there, which coinside with the Did's proof above).