An $L^1$ convergence problem

Question

Is the following true?

If $X_n$ converges almost surely to a non-negative random variable $X$ having finite expectation, and if $E(X_n)$ converges to $E(X)$, then $E|X_n - X|$ converges to $0$?

I am not able to apply the dominated convergence theorem to solve this problem. Any help will be greatly appreciated.

Answer 1

Yes, the statement is true if the random variables are non-negative.

Hint: By the triangle inequality and the positivity of the random variables, we have

$$(X_n+X)-|X_n-X| \geq 0.$$

Now write

$$2 \int X \, d\mathbb{P} = \int \liminf_{n \to \infty} (X_n+X-|X_n-X|) \, d\mathbb{P}$$

and apply Fatous lemma.

Answer 2

As pointed out by saz, it is true if the random variables are non-negative. In general, it is not true: take $X_n$ a random variable taking the value $0$ with probability $1-n^{-2}$, $n^3$ with probability $1/(2n^2)$ and $-n^3$ with probability $1/(2n^2)$. Then $X_n\to 0$ almost surely, $\mathbb E[X_n]=0$ but $\mathbb E|X_n|=n$.