TITLE: Convergence in Distribution of a random variable with an upper bound implies convergence of higher moments.
QUESTION [6 upvotes]: Let $k\in \mathbb{N}$ and $\{X_n\}_{n \in \mathbb{N}}$ be random variables such that they converge in distribution to some $X$ and $|{X_n}|\leq Y$, where $Y$ is a random variable with $\mathbb{E}\left[Y^k\right]<\infty$. I am asked to show that
$$
\mathbb{E}\left[X_n^i\right]\to \mathbb{E}\left[X^i\right]
\quad \forall i=1,2,\dots,k
$$

Now my idea was to just use $Y$ and $Y^i$ as integrable upper bounds and use the dominated convergence theorem. However since convergence in distribution is given, I was wondering if I could use it as well or if I even have to use it.

REPLY [6 votes]: It suffices to prove the claim when $k=1$, the general case will follow by replacing $X_n$ with $(X_n)^i$ and $Y$ with $Y^i$.
Since $\forall n\geq 0$, $|X_n|\leq Y$, the family $(X_n)$ is uniformly integrable. In thay case, it is well-known that $X_n\xrightarrow{d} X$ implies $E(X_n)\to E(X)$.