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

• 成果类型: Article
• 署名作者: Dolgopyat, Dmitry; Hafouta, Yeor
• 署名单位: University System of Maryland; University of Maryland College Park; University System of Ohio; Ohio State University
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-022-01177-2
• 发表日期: 2023
• 页码: 439-476
• 关键词: local limit-theorem; U-statistics; approximation; decay

摘要

We prove a Berry-Esseen theorem and Edgeworth expansions for partial sums of the form S-N = Sigma(N)(n=1) f(n)(X-n, Xn+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 is an element of (0, 1/2). In this case it turns out that expansions of any order r < 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 Holder continuous with some exponent alpha is an element of (0, 1), we show that S-N admits expansions of all orders r <1+alpha/1-alpha. For Holder 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 < 1+alpha/1-alpha is optimal even in the homogeneous case.