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

Aaron Lau, Kouji Yano

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.

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.
Paper Structure (8 sections, 11 theorems, 114 equations)

This paper contains 8 sections, 11 theorems, 114 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Main Results
  4. Proof of WLLN
  5. Proof of SLLN
  6. Construction of splitting chains
  7. Return to small set with geometric tail's probability
  8. The proof of SLLN

Key Result

Theorem 2.4

Suppose Assumption DriftCondition and Assumption ContractionCondition hold. Then there exists a unique sequence $\{\mu_n\}_{n\in \mathbb{Z}}$ of probability measures satisfying for any $n\in \mathbb{Z}$ such that Moreover, the following assertions hold:

Theorems & Definitions (23)

  • Definition 2.1: Liu--Lu LiuLu2025
  • Theorem 2.4: Liu--Lu LiuLu2025
  • Theorem 3.1
  • Theorem 3.3
  • Example 3.4
  • proof : Proof of Theorem \ref{['WLLN']}
  • Remark 5.1
  • Proposition 5.2
  • proof
  • Proposition 5.3
  • ...and 13 more