Law of large numbers for nonnegative random variables

Question

I'm struggling with specific variation of Strong Law of Large Numbers. Suppose $X_1,X_2,\ldots$ are independent, identically distributed, nonnegative random variables and $\mathbb{E} X_1 = \infty $. Show that

$$\mathbb{P}\bigg(\lim_{n\to\infty}\frac{X_1+X_2+\ldots+X_n}{n}=\infty\bigg)=1.$$

Thanks in advance.

Answer 1

For fixed $k \in \mathbb{N}$ set $Y_n := X_n \wedge k$. Then $(Y_n)_{n \in \mathbb{N}}$ is a sequence of iid random variables, $Y_k \in L^1$. By non-negativity and the strong law of large numbers

$$\begin{align*} \liminf_{n \to \infty} \frac{X_1+\ldots+X_n}{n} &\geq \liminf_{n \to \infty} \frac{Y_1+\ldots+Y_n}{n} \\ &= \mathbb{E}(Y_1) \\ &= \mathbb{E}(X_1 \wedge k). \end{align*}$$

Since this holds for any $k \in \mathbb{N}$, we get

$$\liminf_{n \to \infty} \frac{X_1+\ldots+X_n}{n} \geq \sup_{k \geq 1} \mathbb{E}(X_1 \wedge k)= \mathbb{E}(X_1)=\infty.$$