$E(X_n)\rightarrow E(X)\nRightarrow E(|X_n-X|)\rightarrow 0$

Question

If $\{X_n\}$ is a sequence of random variables and $X$ be another random variable, then as far as I know $$E(X_n)\rightarrow E(X)\nRightarrow E(|X_n-X|)\rightarrow 0$$ always. But I am not getting an example. Can somone provide me one?

Answer 1

Let $X_n$ be Rademacher-distributed, i.e. $P(X_n = 1) = P(X_n = -1) = \frac{1}{2}$ and $X = 0$. Then clearly $E[X_n] = 0 \to 0 = E[X]$, but $E[|X_n - X|] = E[|X_n|] = E[1] = 1$.

Answer 2

Well, just pick two different measurable functions $Y$ and $X$ with the same integral and set $X_n=Y$. Then $E[X_n]\equiv E[Y]=E[X]$ but $E[\lvert X_n-X\rvert]\equiv E[\lvert Y-X\rvert]>0$.