# Asymptotic variance for averages of trajectory functionals of Markov chain

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.

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