If a sequence of random variables $X_n$ converges in distribution to some r.v. $X$, the convergence of moments doesn't immediately follow. However, if the sequence is uniformly integrable, then we have the convergence of moments.
Thus, for example, if $X_n\Rightarrow X$ and \sup \mathbb{E}[|X_n|^{1+\varepsilon}]<\infty for some $\varepsilon >0$ (a sufficient condition for uniform integrability), then \mathbb{E}[|X|]<\infty and $\mathbb{E}[X_n]→\mathbb{E}[X]$. (See for example Theorem 25.12 and Corollary in Billingsley's Probability and Measure).
My situation however is this: $X_n\Rightarrow X$, and $\mathbb{E}[X_n]=\infty$.
QUESTION: Does it follow that $\mathbb{E}[X]=\infty$ too?
Let me add that all the $X_n$ and $X$ are nonnegative so their moments are defined ($\mathbb{R}_+ \cup \infty$).
The moment convergence results I've seen all invoke uniform integrability and finiteness of moments, which doesn't apply here. Is it even possible to have \mathbb{E}[X]<\infty (a counterexample would be instructive)? Or might anyone be able to suggest other additional conditions so that $\mathbb{E}[X]=\infty$?