Assume I have a sequence of discrete, nonnegative random variables $X_1,\dots,X_n$ with finite expectation $E[X_n] < \infty$ for all $n \in \mathbb{N}$ that converge in distribution to $X_n\overset{n\to\infty}{\to}X$. I am looking for sufficient conditions on the sequence such that $$ \lim_{n\to\infty} E[X_n] < \infty. $$ Background: Recently I noticed that convergence in distribution does not imply the convergence of expectation. As an example, consider the random variables with $P(X_n=0) = 1-1/n$ and $ P(X_n=n^2) = 1/n$. We directly see that $X_n\to X$ in distribution, where $P(X = 0) = 1$. However, $E[X_n] = n$, which diverges. It seems to me however that that this example is kind of artificial and in many cases convergence could imply convergence of expectation under some mild additional conditions.
On idea is, for example, to limit the support (the largest value $x$ such that $P(X_n=x)>0$) of all $X_n$ to some constant $C$. However, such a constraint is quite restrictive and I am looking for conditions that allow the support to increase with $n$ (or be infinite for all $n$, even).
I am therefore wondering, if someone is aware of such conditions or related literature on that topic.
Edit: It appears that Fatou's Lemma gives at least partially an answer to my question, see, e.g., https://math.stackexchange.com/a/219526/533869
