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

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.