If $\{|X_n|^\beta,\,n\geq1\}$ is uniformly integrable for some $\beta\geq1$ and $S_n=\sum_{i=1}^nX_i$, then $$\left|\frac{S_n}{n}\right|^\beta\mbox{ is uniformly integrable.}$$
$\bullet$ Proof.
$\{|X_n|^\beta,\,n\geq1\}$ is uniformly integrable if and only if the following two conditions are satisfied:
(1) $\mathscr{E}|X_n|^\beta\leq M<\infty, \forall\,n$.
(2) $\forall\,\varepsilon>0,\,\exists\,\delta(\varepsilon)>0$ such that $E\in\mathscr{F}$ with $$\mathscr{P}(E)<\delta(\varepsilon)\implies\int_{E}|x_n|^\beta\,d\mathscr{P}<\varepsilon.$$Next, we check the above two conditions for $\left\{\left|\frac{S_n}{n}\right|^\beta\right\}$.
(1) By Minkowski's inequality, we have $$\begin{array}{rl}
\left(\mathscr{E}\left|\frac{S_n}{n}\right|^\beta\right)^{1/\beta}
& = \frac{1}{n}\left(\mathscr{E}\left|\sum_{i=1}^nX_i\right|^\beta\right)^{1/\beta} \\
& \leq \frac{1}{n}\left\{\left(\mathscr{E}|X_1|^\beta\right)^{1/\beta}+\left(\mathscr{E}\left|\sum_{i=2}^nX_i\right|^\beta\right)^{1/\beta}\right\} \\
& \leq\cdots \\
& \leq \frac{1}{n}\sum_{i=1}^n\left(\mathscr{E}|X_i|^\beta\right)^{1/\beta}\\
& \leq M^{1/\beta} <\infty.\end{array}$$Thus, $$\mathscr{E}\left|\frac{S_n}{n}\right|^\beta\leq M<\infty.$$
(2) [Lack of discussing the event $E$]
$$\begin{array}{rl}
\left(\int_E\left|\frac{S_n}{n}\right|^\beta\,d\mathscr{P}\right)^{1/\beta}
& = \frac{1}{n}\left(\int_E\left|\sum_{i=1}^nx_i\right|^\beta\,d\mathscr{P}\right)^{1/\beta} \\
& \leq \frac{1}{n}\left\{\left(\int_E|x_1|^\beta\,d\mathscr{P}\right)^{1/\beta}+\left(\int_E\left|\sum_{i=2}^nx_i\right|^\beta\,d\mathscr{P}\right)^{1/\beta}\right\} \\
& \leq\cdots \\
& \leq \frac{1}{n}\sum_{i=1}^n\left(\int_E|x_i|^\beta\,d\mathscr{P}\right)^{1/\beta}\\
& \leq \varepsilon^{1/\beta}.\end{array}$$
$\Box$
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