Suppose we have an irreducible and aperiodic discrete-time Markov chain on a finite set $S$ with stationary probability $\pi$; denote it by $(X_0, X_1, X_2,...)$. It is known that for every $x\in S$ and every positive function $f:S\to \mathbb{R}_+$ $$ \frac{1}{n}\sum_{k=0}^{n-1} f(X_k) \to \int f d\pi$$ $\mathbb{P}_x$ almost surely. Are there any good resources where the speed of this convergence, namely, the quantity (for an $f\in L^1(\pi)$) $$ \mathbb{P}_x\bigg( \bigg|\frac{1}{n}\sum_{k=0}^{n-1} f(X_k) -\int f d\pi \bigg| > \varepsilon\bigg)$$ is explored ? I guess this probability decays exponentially for every $x\in S$ in the "ideal" (finite, irreducible, aperiodic) case. Are there any books that answer this question in this simple setting ? I am not looking for generalisations to arbitrary spaces, that most recent papers deal with, but rather a simple, clean and concise statement and proof !
More precisely, I am interested in the case where $f(x)=f_y(x)=1\{x=y\}$ for $y \in S$, since I would like to have exact quantitative bounds on the probability that the empirical measure of the Markov chain deviates from the true invariant probability by more than some small amount.