# $X_1,X_2, \ldots$ be i.i.d. Show that $\mathbb{E}|X_1| < \infty$ iff $\frac{X_n}{n} \to 0$ a.s

Suppose $$X_1,X_2, \ldots$$ be i.i.d. Show that $$\mathbb{E}|X_1| < \infty \Leftrightarrow \frac{X_n}{n} \to 0$$ a.s

I tried using Markov but I don't know anything about the $$\mathbb{E}X$$. I was also thinking of borel Cantelli, to show $$\sum P\left[\frac{X_n}{n}>\varepsilon\right]< \infty$$ for each $$\varepsilon>0$$, then invoke First Borel Cantelli but I am confused on how to even get to the sum is less than infinity part

This is easily proved using Borel-Cantelli Lemma and the following well known fact:

For a non-negative random variable $$Y$$ we have $$EY <\infty$$ iff $$\sum P(Y>n) <\infty$$.

Now let $$\epsilon >0$$. Taking $$Y=\frac {|X_1|} \epsilon$$ we see that $$E|X_1|<\infty$$ iff $$\frac{E|X_1|}{\epsilon}<\infty$$ iff $$\sum P\left(\frac {|X_1|} {\epsilon} >n\right) <\infty$$ iff $$\sum P\left(\frac {|X_n|} n >\epsilon\right) <\infty$$ iff $$P\left( \frac {|X_n|} n >\epsilon\hspace{0.2cm} \text{ i.o.}\right)=0$$. Can you finish the proof?

• Perfect!! thank you so much Jun 1, 2020 at 0:38
• Can we somehow connect this a new question Q . For any sequence of r.v $X_n$, show that there exists constants $c_n \to \infty$ s.t $X_n/c_n \to 0$ a.s Jun 1, 2020 at 0:41
• @foobar Choose $c_n$ such that $P(|X_n| >\frac {c_n} n) <\frac 1 {n^{2}}$ and apply Borel-Cantelli. Jun 1, 2020 at 5:23

Another solution is this:

Let $$S_n=X_1+...+X_n$$

Note that $$\frac{X_n}{n}=\frac{S_n}{n}-\frac{S_{n-1}}{n-1} \left(\frac{n-1}{n}\right)$$

Then use the strong law of large numbers.