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I am looking for references on theory for convergence rates of ergodic averages of a Markov chain in the more general setting where the functional is over multiple states or even a whole trajectory, but I am struggling to find the right keywords or even instances of these problems being considered.

Let $(X_n)_{n \in \mathbb N_0}$ be a Markov chain on a general state space $(\mathsf X, \mathscr B(\mathsf X))$, and let $\pi$ be an invariant distribution. There are several approaches to establishing a Markov chain law of large numbers; one of them is Birkhoff's ergodic theorem, whence under some mixing conditions one forms a dynamical system of paths of the Markov chain and obtains $\frac 1n\sum_{k=0}^{n-1} f(X_k) \to \mathbb{E}_{\pi}[f]$ a.s. and in $L^1$ for $\pi$-a.e. $X_0$. Many presentations stop there, but in fact the theorem holds for functionals of paths, so that $$\frac 1n \sum_{k=0}^{n-1} F(X_k, X_{k+1}, \dotsc) \xrightarrow[n \to \infty]{\text{a.s.}, L^1} \mathbb{E}_{\mathbb{P}_\pi}[F].$$

Now we might want to quantify the rate of this convergence, for example by the asymptotic variance $$\lim_{n \to \infty} n \mathrm{Var}_{\mathbb{P}_\pi}\left(\frac 1n \sum_{k=0}^{n-1} F(X_k, X_{k+1}, \dotsc)\right).$$ Question: Under what conditions on the chain and $F$ is this limit finite?

My setting has a "nice" geometrically ergodic Markov chain, and so I am interested even in quite strong conditions, but it is not reversible. Fairly simple sufficient conditions have been developed in connection to the CLT in the "usual" setting where $F(X_k)$ only depends on the current state, see e.g. these notes by Eberle (2023) or this survey by Jones (2004). I have found a slight generalization in Jensen (1989) for the case $F(X_k,X_{k+1})$. For "finite horizons" i.e. $F(X_k,X_{k+1},\dotsc,X_{k+\ell})$ one might hope to connect to theory for strongly mixing sequences as in this question; there is a well-known AR(1) counterexample where strong mixing fails with infinite horizons, although for my purpose it would suffice with finite (but not independent) random horizon.

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Obviously asking the question caused my literature search to uncover exactly the right reference. Essentially the common hypotheses as in the "usual" setting suffice for a finite horizon, but remarkably also if one has a stopping time horizon (making it a finite but random horizon).

Ben Alaya, M., & Pagès, G. (1998). Rate of Convergence for Computing Expectations of Stopping Functionals of an $\alpha$-Mixing Process. Advances in Applied Probability, 30(2), 425–448. http://www.jstor.org/stable/1427976

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