发表机构
University of Florida; Ben-Gurion University of the Negev（佛罗里达大学; 内盖夫本-古里安大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对局部平稳马尔可夫链的部分和，证明了Berry--Esseen定理、局部中心极限定理、Edgeworth展开及大偏差原理，并给出渐近均值和方差的显式公式及最优收敛速率。
AI 中文摘要
局部平稳马尔可夫链在过去三十年中引起了统计学界的广泛关注。本文证明了由这类链生成的部分和的多种极限定理。我们首先给出了渐近均值和方差（以及更高阶矩）的显式公式，并获得了向它们收敛的最优速率。随后，我们证明了Berry--Esseen定理、局部中心极限定理、Edgeworth展开以及大偏差和中偏差原理。我们的方法涉及一个参数化的Perron--Frobenius定理，该定理利用\\(\cite{Rugh,Dubois}\\)中发展的复（希尔伯特）射影度量理论，结合局部逼近论证及\\(\cite{dolgopyat2023berry}\\)中的思想来证明。
英文摘要
Locally stationary Markov chains have attracted a lot of attention in statistics in the past three decades. In this paper we prove a variety of limit theorems for partial sums generated by such chains. We first provide explicit formulas for the asymptotic mean and variance (and also higher moments), and obtain optimal convergence rates towards them. We then prove a Berry--Esseen theorem, a local central limit theorem, Edgeworth expansions, and large and moderate deviations principles. Our approach involves a parametric Perron--Frobenius theorem, which is proved using the theory of complex (Hilbert) projective metrics developed in \cite{Rugh,Dubois}, together with local approximation arguments and ideas in \cite{dolgopyat2023berry}.
Comments28 pp
