I am finding two results online and I was wondering which is true.
Let $(M_n)$ be random variables such that $M_n\to_d M$ for some random variable $M$. If $(|M_n|^p)$ is uniformly integrable for some $p>1$, then $\mathbb{E}[|M_n|^q]\to \mathbb{E}[|M|^q]$ for all $0<q<p$.
However, in some places it holds for $0<q\le p$. Which is the correct one?!
Regarding the first statement ($q<p$), this seems clear - if $q<p$, then $|M_n|^q$ is $L^{p/q}$ bounded, hence UI. By Skorokhod theorem and Vitali's theorem, the result follows. But this argument does not work when $q=p$...
Thanks!
For the case $p=q$: replacing $M_n$ by $\lvert M_\rvert^p$ if needed, we can assume that $M_n\geqslant 0$ almost surely and that $p=1$. From the formula $\mathbb E\left[X\right]=\int_0^{+\infty}\Pr\{X>t\}dt$, we derive that for all fixed $R$, $$ \left\lvert \mathbb E\left[M_n\right]-\mathbb E\left[M\right] \right\rvert \leqslant \int_0^R\left\lvert \Pr\{M_n>t\}-\Pr\{M>t\}\right\rvert dt +\int_R^{+\infty}\Pr\{M_n>t\}dt+\int_R^{+\infty}\Pr\{M >t\}dt. $$ From the uniform integrability assumption, the second term of the right hand side goes to $0$ uniformly with respect to $n$ as $R$ goes to infinity. For a fixed $R$, the first term goes to zero as $n$ goes to infinity by the dominated convergence theorem.