TITLE: Convergence of moments of a sequence of random variables
QUESTION [5 upvotes]: I encountered this problem in my study of time series. It seemed trivial at first but I don't see the finishing move to complete the proof. The problem is as follows.
Let $(X_n)_n$ be a sequence of random variables and $X$ also a random variable on some probability space. Suppose that $E\left[\lvert X_n - X\rvert^p\right]\to 0$ as $n$ goes to infinity and $E\left[\lvert X\rvert^p\right]< \infty$ for some $p \geq 1$. Show that $E[X_n^k] \to E[X^k]$ for every $0<k\leq p$.
We of course know that $E\left[\lvert X_n - X\rvert^p\right]\to 0$ implies $E\left[\lvert X_n - X\rvert^k\right]\to 0$ for every $0<k\leq p$ (by Jensen). By triangle inequality, $E\left[\lvert X_n - X\rvert^k\right]\to 0$ implies $E\left[\lvert X_n \rvert^k\right]\to E\left[\lvert X\rvert^k\right]$. I have not made use of the facts that $E\left[\lvert X\rvert^p\right]< \infty$ and $p\geq 1$ so far. I also find it strange that $k$ is not restricted to positive integers in the statement of the problem (suppose $X = -1$ a.s. and $k$ is irrational.)

REPLY [3 votes]: This follows from a generalized dominated convergence theorem. Let $\zeta_n:=X_n^k$ and $\zeta:=X^k$. Since $\mathbb{E}|\zeta_n|\le c_k[\mathbb{E}|X_n-X|^k+\mathbb{E}|\zeta|]$ and $k\le p$, $\zeta_n\in L^1$ ($\zeta\in L^1$ by assumption). In addition, you've already shown that $\mathbb{E}|\zeta_n|\to\mathbb{E}|\zeta|$. Thus, for any subsequence $n_m$, there is a further subsequence $n_m'$, s.t. $\zeta_{n_m'}\to\zeta$ a.e. and by the DCT
$$\lim\int \zeta_{n_m'}dP=\int \zeta dP$$