$X_1, ..., X_n$ are i.i.d. and $E(|X_i|)$ is finite. How to show that $(|X_1|+\cdots+|X_n|)/n$ converges to $E(|X_i|)$ in probability?

Question

$X_1, \cdots, X_n$ are i.i.d. and $E(|X_i|)$ is finite.

Show that $(|X_1|+\cdots+|X_n|)/n$ converge to $E(|X_i|)$ in probability and that $E((|X_1|+\cdots+|X_n|)/n)$ converge to $E(|X_i|)$.

I think this may need characteristic functions or law of large numbers to solve but don't know how. Can someone give some hint?

Answer 1

If $\{X_i\}$ is an i.i.d sequence of random variables, then so is $\{|X_i|\}$. Now you can apply the weak law of large numbers to the sequence $\{|X_i|\}$.

Answer 2

The second half seems trivial.

$$ \mathbb{E}( (|X_1| + |X_2| + \dots) /n ) = \frac{1}{n}(\mathbb{E}(|X_1|)+\mathbb{E}(|X_2|)+\dots) = \mathbb{E}(|X_1|)$$

as they are identically distributed. (We don't even need independence).

I guess this is convergence but this is just a constant sequence.