# Quantitative version of ergodic theorem in Markov chains

Consider an irreducible Markov chain $$X_t$$ with finite state space $$E$$, and unique invariant measure $$\pi$$. Fix a function $$V:E\to\mathbb R$$ such that $$E_\pi[V]=0$$. The ergodic theorem tells us that, almost surely, $$\frac{1}{t}\int_0^t V(X_s)ds \to E_\pi[V]=0.$$

My question is, is there an estimate on the rate of convergence in $$L^p$$(at least for $$p=1,2$$)? Namely, can we bound the $$L^p$$ norm of $$\frac{1}{t}\int_0^t V(X_s)ds$$ in terms of $$C(V)t^{-m}$$? where $$m$$ is some constant related to the mixing property of the chain $$X_t$$, like mixing time or spectral gap.

At least in $$L^2$$ this question has been studied in the context of Markov central limit theorems. Let $$\mathbb P_\pi$$ be the induced measure on path space when starting according to $$\pi$$. We have $$\mathrm{Var}_{\mathbb P_\pi}\left( \frac 1t \int_0^t V(X_s)\,ds \right) = \frac 2t \int_0^t \left(1 - \frac rt\right)\mathrm{Cov}_{\mathbb P_\pi}[V(X_0),V(X_r)]\,dr$$ Now one needs to obtain fast enough decaying bounds for the covariances, to obtain something like $$\le C/t$$ as suggested in your question. Since your state space is finite you will hopefully be able to obtain quite strong mixing results.