I have a question as follows: we know that convergence in distribution does not imply convergence of moments. In fact by Fatou's lemma and Skorohod's representation, if X_n -> X in distribution, we have

liminf E[X_n^2] >= E[X^2]

and strict inequality is possible, e.g., consider P{X_n = 0} = 1-1/n and P{X_n = n} = 1/n, X_n -> 0 in distribution.

Now, how about variance? My question is that is it possible to construct an example such that X_n -> X in distribution and

liminf var X_n < var X < Infinity?

Clearly this could happen only when E[X_n^2] is unbounded.