0.1 Convergence of Moments
Let \(X_{n}\stackrel{\mathcal{D}}{\to}X\), then for all \(\beta \in (0,\infty)\) the following are equivalent: 1. \(\mathbb{E}[\lvert X_{n} \rvert^\beta]< \infty\) for all \(n \geq 1\), \(\mathbb{E}[\lvert X \rvert^\beta]<\infty\), \(\mathbb{E}[\lvert X_{n} \rvert^\beta]\to \mathbb{E}[\lvert X \rvert^\beta]\). 2. \(\{ \lvert X_{n} \rvert ^\beta \}_{n \geq 1}\) is uniformly integrable.
Note: It is enough to require that there exists \(\alpha > \beta\) such that \(\sup_{n \geq 1}\mathbb{E}[\lvert X_{n} \rvert^\alpha]< \infty\) to have uniform integrability.
\begin{proof} For a proof of this result see [[files-notes-pstat213bc-lecture-notes-raya-pdf|PSTAT213BC Lecture Notes (Raya), page 43]] \end{proof}