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

Abstract

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 a uniformly elliptic inhomogeneous Markov chain and \(\{f_n\}\) is a sequence of uniformly bounded functions. The Berry–Esseen theorem holds without additional assumptions, while expansions of order 1 hold when \(\{f_n\}\) 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 \(f_n\) is of order \(O(n^{-{\beta }})\) for some \({\beta }\in (0,1/2)\). In this case it turns out that expansions of any order \(r<\frac{1}{1-2{\beta }}\) hold, and this condition is optimal. The second case is uniformly elliptic chains on a compact Riemannian manifold. When \(f_n\) are uniformly Lipschitz continuous we show that \(S_N\) admits expansions of all orders. When \(f_n\) are uniformly Hölder continuous with some exponent \({\alpha }\in (0,1)\), we show that \(S_N\) admits expansions of all orders \(r<\frac{1+{\alpha }}{1-{\alpha }}\). For Hölder continues functions with \({\alpha }<1\) our results are new also for uniformly elliptic homogeneous Markov chains and a single functional \(f=f_n\). In fact, we show that the condition \(r<\frac{1+{\alpha }}{1-{\alpha }}\) is optimal even in the homogeneous case.