Show $(X_n)$ is a sequence of i.i.d. random variables such that $S_n/n\to -\infty$

Question

$(X_n)_{n\geq 1}$ is a sequence of i.i.d. random variables such that $E{X_n}^+<\infty$ and $E{X_n}^-=\infty$. I want to show $\frac{S_n}{n}\to -\infty$ almost surely. The problem is that I cannot apply SLLN since $E|X_1|=\infty$. My immediate thought is that I have to construct a new sequence of i.i.d. random variables from the original sequence apply SLLN to this sequence and relate it back to what happens to the original sequence. The problem is, I am having trouble finding a good candidate sequence. Any hints would be appreciated.

Answer 1

Since $X_i=X_i^+-X_i^-$, the expression of $S_n/n$ is $n^{-1}\sum_{i=1}^nX_i^+-n^{-1}\sum_{i=1}^nX_i^-$. By the usual strong law of large numbers, $n^{-1}\sum_{i=1}^nX_i^+\to\mathbb E\left[X_1^+\right]$ which is finite. Therefore, it suffices to prove that if $(Y_i)$ is an i.i.d. sequence where $\mathbb E\left[Y_1\right]=\infty$, then $n^{-1}\sum_{i=1}^nY_i\to \infty$ almost surely.

To do so, for each integer $M$, define $Y_{i,M}:=Y_i\mathbf{1}_{\{Y_i\leqslant M\}}$. Then $n^{-1}\sum_{i=1}^nY_i\geqslant n^{-1}\sum_{i=1}^n Y_{i,M}$ hence taking the $\liminf_{n\to\infty}$ and using the stong law of large numbers, $$ \liminf_{n\to\infty} n^{-1}\sum_{i=1}^nY_i\geqslant\liminf_{n\to\infty} n^{-1}\sum_{i=1}^n Y_{i,M}=\mathbb E\left[Y_1\mathbf{1}_{\{Y_1\leqslant M\}}\right] \mbox{a.s.}. $$ We conclude by letting $M$ going to infinity and using the monotone convergence theorem.