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Berry--Esseen theorems, LLT, Edgeworth expansions, and large deviations for Locally Stationary Markov Chains

  • Yeor Hafouta
  • Brendan Williams
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Abstract

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}.

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Cite

BibTeXRIS
BibTeX
@misc{hafouta2026berryesseen,
  title = {{Berry--Esseen theorems, LLT, Edgeworth expansions, and large deviations for Locally Stationary Markov Chains}},
  author = {Yeor Hafouta and Brendan Williams},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.33435},
  eprint = {2609.33435},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.33435},
}
RIS
TY  - UNPB
TI  - Berry--Esseen theorems, LLT, Edgeworth expansions, and large deviations for Locally Stationary Markov Chains
AU  - Yeor Hafouta
AU  - Brendan Williams
PY  - 2026
DA  - 2026/09/27
PB  - arXiv
DO  - 10.48550/arxiv.2609.33435
UR  - https://arxiv.org/abs/2609.33435
AB  - 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}.
ER  -