A Berry–Esseen theorem and Edgeworth expansions for uniformly elliptic inhomogeneous Markov chains

We prove a Berry–Esseen theorem and Edgeworth expansions for partial sums of the form $$\displaystyle S_N=\sum \nolimits _{n=1}^{N}f_n(X_n,X_{n+1})$$ , where $$\{X_n\}$$ is uniformly elliptic inhomogeneous Markov chain $$\{f_n\}$$ sequence bounded functions. The holds without additional assumptions, while order 1 hold when irreducible, which an optimal condition. For higher expansions, we then focus on two situations. first essential supremum $$f_n$$ $$O(n^{-{\beta }})$$ some $${\beta }\in (0,1/2)$$ . In this case it turns out that any $$r<\frac{1}{1-2{\beta }}$$ hold, condition optimal. second chains compact Riemannian manifold. When are Lipschitz continuous show $$S_N$$ admits all orders. Hölder with exponent $${\alpha (0,1)$$ orders $$r<\frac{1+{\alpha }}{1-{\alpha continues functions }<1$$ our results new also homogeneous single functional $$f=f_n$$ fact, even in case.