Markov chains in the domain of attraction of Brownian motion in cones

TL;DR

This paper studies multidimensional Markov chains converging to Brownian motion within cones, constructing harmonic functions to analyze their exit times and asymptotic behaviors.

Contribution

It introduces a method to construct harmonic functions for Markov chains in cones, linking their asymptotic behavior to that of Brownian motion.

Findings

Harmonic functions for Markov chains in cones are similar to those of Brownian motion.

Asymptotic tail behavior of exit times is characterized using these harmonic functions.

The approach provides insights into the probabilistic structure of Markov chains near boundaries.

Abstract

We consider a multidimensional Markov Chain $X$ converging to a multidimensional Brownian Motion. We construct a positive harmonic function for $X$ killed on exiting the cone. We show that its asymptotic behavior is similar to that of to the harmonic function of Brownian motion. We use the harmonic function to study the asymptotic behaviour of the tail distribution of the exit time $\tau$ of $X$ from a cone.