The Law of Large Numbers for Time-inhomogeneous Markov Chains under General Conditions

TL;DR

This paper establishes weak and strong laws of large numbers for time-inhomogeneous Markov chains under broad conditions, extending classical ergodic results to more general, non-stationary settings.

Contribution

It introduces new conditions and methods to prove LLNs for time-inhomogeneous Markov chains, including a Nummelin-type splitting under minimal assumptions.

Findings

Proved weak LLN under Drift and Contraction Conditions.

Established strong LLN using Doeblin minorization and splitting.

Extended Harris-ergodic LLN to time-inhomogeneous chains.

Abstract

The weak and strong laws of large numbers for time-inhomogeneous Markov chains are studied under general conditions. First, under Drift Condition and Contraction Condition in total variation, we prove the weak law of large numbers. Then, assuming Drift Condition together with a time-inhomogeneous Doeblin minorization, we develop a Nummelin-type splitting and obtain a strong law of large numbers. Our results utilize the invariant measure family in the sense of Liu--Lu (2025), and extend the classical Harris-ergodic LLN to the time-inhomogeneous setting.