Given an ergodic Markov chain $(X_n)_{n\geq 1}$ in $R^d$with $\pi$ as the invariant distribution of the transition kernel, under good conditions we have that the empirical occupation measure converges to $\pi$, i.e.:

\begin{align} \forall A\in \mathcal{B}(R^d), S^A_N = \frac{1}{N}\sum_{k=1}^{N}1_{X_k \in A} \to \pi(A), \text{ as } N \to +\infty \end{align}

We can even establish a CTL theorem for this convergence: $\sqrt{N}(S_N^A - \pi(A))$ converges to a centered normal distirbution

That's about the asymptotic behavoir. I am wondering if there exsits some non-asymptotic control of the convergence rate. in $L^p$ for example, if there any reference about controlling $||S_N^A - \pi(A)||_{L^p}$?

Thank you for any help!