Induction of Markov chains, drift functions and application to the LLN, the CLT and the LIL with a random walk on $\mathbb{R}_+$ as an example

TL;DR

This paper develops a framework linking invariant measures, drift functions, and Poisson's equation solutions for Markov chains, applying it to establish limit theorems for a specific random walk on the positive real line.

Contribution

It introduces a bijection between invariant measures of original and induced Markov chains and uses drift functions to analyze excursions and limit laws for the walk.

Findings

Established a bijection between invariant measures of original and induced chains.

Linked solutions of Poisson's equation for both chains using drift functions.

Proved LLN, CLT, and LIL for the specific random walk on .

Abstract

Let $(X_n)$ be a Markov chain on a standard borelian space $\mathbb{X}$. Any stopping time $\tau$ such that $\mathbb{E}_x\tau$ is finite for all $x\in\mathbb{X}$ induces a Markov chain in $\mathbb{X}$. In this article, we show that there is a bijection between the invariant measures for the original chain and for the induced one. We then study drift functions and prove a few relations that link the Markov operator for the original chain and for the induced one. The aim is to use this drift function and the induced operator to link the solution to Poisson's equation $(I\_d-P)g=f$ for the original chain and for the induced one. We also see how drift functions can be used to control excursions of the walk and to obtain the law of large numbers, the central limit theorem and the law of the iterated logarithm for martingales. We use this technique to study the random walk on $\mathbb{R}_+$ defined by $X_{n+1} = \max( X_n + Y_{n+1}, 0)$ where $(Y_n)$ is an iid sequence of law $\rho^{\otimes \mathbb{N}}$ for a probability measure $\rho$ having a finite first moment and a negative drift.