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!



convergence rate of occupation measure of ergodic Markov Chain
Here is a finite time bound on $||S\_N-\pi||\_1$:
<https://dl.dropboxusercontent.com/u/3198145/chernoff-dkw.pdf>
Theorem 1.3


