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 SN = N ∑ n=1 fn(Xn, Xn+1), where {Xn} is a uniformly elliptic inhomogeneous Markov chain and {fn} is a sequence of uniformly bounded functions. The Berry-Esseen theorem holds without additional assumptions, while expansions of order 1 hold when {fn} is irreducible, which is an optimal condition. For higher order expansions, we then focus on two situations. The first is when the essential supremum of fn is of order O(n ) for some β ∈ (0, 1/2). In this case it turns out that expansions of any order r < 1 1−2β hold, and this condition is optimal. The second case is uniformly elliptic chains on a compact Riemannian manifold. When fn are uniformly Lipschitz continuous we show that SN admits expansions of all orders. When fn are uniformly Hölder continuous with some exponent α ∈ (0, 1), we show that SN admits expansions of all orders r < 1+α 1−α . For Hölder continues functions with α < 1 our results are new also for uniformly elliptic homogeneous Markov chains and a single functional f = fn. In fact, we show that the condition r < 1+α 1−α is optimal even in the homogeneous case.