My question kind of generalizes the question in Analogue of continuous mapping theorem for convergence in $L_2$ and it is related to the answer by Nate Eldredge.
Edited to give more details:
Suppose that
\begin{equation}\tag{1}\label{eq1}
E|X_n - X|^p = O(b_{p,n}),
\end{equation}
where $p \geq 1$ and $b_{p,n} \to 0$ as $n \to \infty$. Furthermore, assume that there exist constants $0 < a < b < \infty$ such that $a \leq X_n \leq b $ a.e. and $a \leq X \leq b$ a.e.
Now, let $g$ be a continuous function with the following property: there exists a constant $C>0$ where $\sup_{x\in(a,b)}|g(x)| < C$. For example, $g(x) = x^r$, $r>0$, is a possible function. In this case, what can I say about $E|g(X_n) - g(X)|^p$, $p\geq1$? Is it possible to find a rate of convergence based on \eqref{eq1} and the above mentioned assumptions?
In a more specific case, is it possible to find the rate of convergence for $E|X_n^r - X^r|^p$, with $r > 0$ and $p \geq 1$, assuming \eqref{eq1} and that $a < X_n < b$ a.e. and $a<X<b$ a.e.?
Since $a < X_n < b$ a.e. and $a<X<b$ a.e., I think that I can assume, as discussed by Nate, that $L_p$ convergence is guaranteed (we have that $g(X_n) \to g(X)$ in measure by the continuous mapping theorem, and in $L_p$ by the dominated convergence theorem), but I could not find any rate of convergence in my problem.
Thanks in advance.
Since a continuous function on a compact set is Lipschitz, $$ \mathsf{E}|g(X_n)-g(X)|^p\le L^p\mathsf{E}|X_n-X|^p=O(b_{p,n}), $$ where $L$ is the corresponding Lipschitz constant.