Reference for finite-state Markov chain Edgeworth expansion

Identify a published reference that rigorously establishes a first‑order Edgeworth expansion for partial sums of an integer‑valued function of a stationary, irreducible, aperiodic finite‑state Markov chain, in a form directly applicable to the setting of Theorems 3.2 and 3.4 (including the discrete correction term and uniform error bounds).

Background

The authors need an Edgeworth expansion tailored to lattice‑valued sums arising from functions of finite‑state Markov chains. While related results exist for Markov chains and for non‑lattice settings, they report not finding a general theorem that directly applies to their discrete case and therefore supply Theorems 3.2 and 3.4 with full proofs.

They note their result may not be new but lack a citation, making the existence of a suitable prior reference an explicit unresolved item.