I am working on an exercise where I am trying to show that the moments of a $t$ distribution converge in probability to the moments of a standard normal. I'm in need of help with what I think is the trick of the solution, namely uniform integrability.
The question is this:
If $X_n\sim t(n)$, then $X_n \to_d N(0, 1)$. Does it follow that $EX_n^p \to EN(0, 1)^p$ for every $p\in\mathbb N$? Is this true?
I know convergence in distribution does not imply convergence in probability of the moments, but I haven't done anything more formal than that for the first question. For the second, I'm guessing it is true and that I have to use asymptotic uniform integrability, due to this theorem:
Theorem. Let $f: \mathbb R^k\mapsto\mathbb R$ be measurable and continuous at every point in a set $C$. Let $X_n\to_d X$ where $X$ takes its values in $C$. Then $Ef(X_n)\to Ef(X)$ if and only if the sequence of random variables $f(X_n)$ is asymptotically uniformly integrable.
So I'm pretty sure I have to show that $X^p_n$ is uniformly integrable asymptotically, but I am not quite certain how to do that. In particular, how do I do that for all $p$?