[Paper Review] Local limit theorems for inhomogeneous Markov chains

Dmitry Dolgopyat ans Omri Sarig|arXiv (Cornell University)|Sep 12, 2021

Stochastic processes and statistical mechanics96 references4 citations

TL;DR

This paper establishes Local Limit Theorems for bounded additive functionals of uniformly elliptic inhomogeneous Markov chains using novel reduction theorems for Markov arrays. It derives precise asymptotics in the large deviation regime, providing sharp probabilistic estimates for functionals of such chains.

ABSTRACT

We prove the Local Limit Theorems for bounded additive functionals of uniformly elliptic inhomogeneous Markov arrays. As an application we obtain the precise asymptotics in the large deviation regime for bounded additive functionals of uniformly elliptic Markov chains. The proofs rely on new reduction theorems for Markov arrays.

Motivation & Objective

• To develop local limit theorems for bounded additive functionals of uniformly elliptic inhomogeneous Markov chains.
• To establish precise asymptotics in the large deviation regime for such functionals.
• To introduce and apply new reduction theorems for Markov arrays to handle inhomogeneity and uniform ellipticity.
• To extend classical limit theorems to non-stationary, inhomogeneous Markov settings with bounded additive functionals.

Proposed method

• The authors introduce new reduction theorems that simplify the analysis of inhomogeneous Markov arrays by reducing them to more tractable forms.
• They apply spectral and coupling techniques to control the transition kernels under the uniform ellipticity condition.
• The proof framework leverages martingale approximations and characteristic function methods to derive local limit behavior.
• The analysis is conducted under the assumption of bounded additive functionals, ensuring moment control and regularity.
• Reduction theorems are used to decouple time-inhomogeneity and reduce the problem to a sequence of homogeneous-like components.
• The method enables precise asymptotic control in the large deviation regime by combining local limit results with exponential moment estimates.

Experimental results

Research questions

• RQ1 How can local limit theorems be extended to inhomogeneous Markov chains with bounded additive functionals?
• RQ2 What are the precise asymptotics of additive functionals in the large deviation regime for uniformly elliptic inhomogeneous Markov chains?
• RQ3 What structural reductions can be applied to inhomogeneous Markov arrays to enable limit theorem derivations?
• RQ4 How does uniform ellipticity facilitate the derivation of sharp asymptotic bounds in non-stationary settings?
• RQ5 Can new reduction theorems for Markov arrays provide a general framework for analyzing inhomogeneous Markov processes?

Key findings

• The paper establishes local limit theorems for bounded additive functionals of uniformly elliptic inhomogeneous Markov chains.
• It derives precise asymptotics in the large deviation regime for such functionals, providing sharp tail estimates.
• The new reduction theorems for Markov arrays are instrumental in handling time-inhomogeneity and enabling the limit theorems.
• The results hold under the condition of uniform ellipticity, ensuring sufficient mixing and regularity of transitions.
• The framework allows for quantitative control of probabilities in the large deviation regime via local limit behavior.
• The approach generalizes classical results to non-stationary settings while maintaining sharp asymptotic precision.