Convergence in Distribution, Expectation, Variance

Question

Is there a sequence of random variables $X_n$ converging to some $X$ in distribution with neither $\mathbb{E}[X_n] \to \mathbb{E}[X]$ nor $\mathbb{V}[X_n] \to \mathbb{V}[X]$?

Answer 1

Yes, just take any random variable $X$ such that $\mathbb{E}(|X|) = \infty$, then $X_n := X$ converges to $X$ in distribution, but neither $\mathbb{E}(X_n)$ nor $\mathbb{V}(X_n)$ are well-defined, and so mean and variance cannot converge.

Even if $X_n$ and $X$ are nicely integrable, we can, in general, not expect to have convergence in $L^1$ or $L^2$. Consider, for instance,

$$X_n(\omega) := n 1_{(0,1/n)}(\omega)$$

on $\Omega := (0,1)$ endowed with the Lebesgue measure. Then $X_n \to 0$ almost surely (hence in distribution), but you can easily check that neither $\mathbb{E}(X_n)$ nor $\mathbb{V}(X_n)$ converges to $0$.