Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2609.33435

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

Yeor Hafouta·Brendan Williams
View original source ↗Read source document ↗Export metadata ↓

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}.

Explore related subjects

math.PRmath.STstat.TH

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Yeor Hafouta, Brendan Williams. 2026-09-27. Berry--Esseen theorems, LLT, Edgeworth expansions, and large deviations for Locally Stationary Markov Chains. https://arxiv.org/abs/2609.33435

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Permuton and local limits for the Luce model

We investigate the asymptotic properties of permutations drawn from the Luce model, a natural probabilistic framework in which permutations are generated sequentially by sampling without replacement, with selection probabilities proportional to prescribed positive weights. These permutations arise in applications such as ranking models, the Tsetlin library, and related Markov processes. Under minimal assumptions on the weights, we establish a permuton limit theorem describing the global behavior of Luce-distributed permutations and derive an explicit density of the limiting permuton. We further compute limiting pattern densities and analyze the differences between exact Luce permutations and their permuton approximations. We also study the local convergence of these permutations, proving a quenched Benjamini--Schramm limit and a central limit theorem for consecutive pattern occurrences. Finally, we prove a central limit theorem for the number of inversions.

math.PR↗
arXiv2026

A Sharper Hoeffding Bound for Weighted Sums of Exchangeable Random Variables

We prove a Hoeffding-type moment generating function bound for weighted sums of bounded exchangeable random variables centered by their finite-population average. The bound reduces the excess inflation above one in a recent weighted exchangeable Hoeffding inequality from order $(\log N)/N$ to the rate-optimal order $1/N$, with an explicit constant. The proof reduces the problem to Hamming slices, identifies two-level extremizers for the relevant symmetric variational problem, and applies a hypergeometric martingale bound. We also give a lower bound showing that an excess inflation of order $1/N$ is unavoidable.

math.PR↗
arXiv2026

Angle Distributions for Intersecting Random Segments in Star-Shaped Planar Domains

Let $Ω\subset\mathbb{R}^2$ be a bounded planar set that is star-shaped with respect to the origin, and let $A,B,C,D$ be independent random points uniformly distributed on $Ω$. We consider the random segments $S_{AB}$ and $S_{CD}$ and study the distribution of the smaller angle $Θ\in[0,π/2]$ formed by them, conditional on the event that they intersect. Using the radial function of $Ω$, together with a parametrization of each segment in terms of its supporting line and the positions of its endpoints along that line, we derive an integral representation for the conditional distribution \[ \Pr\{Θ\leqθ| S_{AB}\cap S_{CD}\neq\varnothing\}. \] The resulting expression makes explicit how the geometry of the boundary of $Ω$ determines the angular distribution. The probability of intersection appears naturally as the normalizing constant and is related to the probabilistic version of Sylvester's four-point problem.

math.PR↗