Prove that $\sum_{j=1}^{n} X_{j}$/ $E(\sum_{j=1}^{n} X_{j})$ converges a.c.

Question

I would like some help proving the following result. Thanks for any help in advance.

Let $(X_n)_{n=1}^\infty$ be a sequence of nonnegative $\ell_{1}$ random variables with $EX_{n} > 0, n \geq 1$. Set $S_{n} =\sum_{j=1}^{n} X_{j}, n \geq 1$ Prove that if

$\sum_{n=1}^{\infty} (EX_{n}/ES_{n}) < \infty$,

then $S_{n}/ES_{n}$ converges a.c.

Hint. Consider the two cases $\sum_{j=1}^{\infty} EX_{j} < \infty$ and $\sum_{j=1}^{\infty} EX_{j} = \infty$

I am unsure how to begin

Edit 1: I think that as a first step, I should apply Kronecker's Lemma.

Edit 2:I'm also wondering if I should apply the Beppo Levi Theorem.

Edit 3: I believe that I should only apply Kronecker's lemma in the second case, where the infinite sum of the expectations equals infinity.

Edit 4: I think I have solved the case where the sum of the expectations is finite. I only need to work on the infinite case.

Answer 1

$S_n$ is increasing and $E[S_n]=E[\sum_{j \le n} X_j]=\sum_{j\le n} E[X_j]$, so that $E[S_n]\uparrow E[\lim S_n]=E[\sum X_j]=\infty$.

Then by Kronecker's Lemma, we can know that $\frac{\sum_{j\le n} E[X_j]}{E[S_n]} \to 0$, but $\frac{\sum_{j \le n} E[X_j]}{E[S_n]} = 1,\forall n$. This would lead to a contradiction.