Limiting Markov Chain Distributions

Basic Concepts

A distribution π of a Markov chain, as defined in Markov Chain Distributions, is a limiting distribution if for all states i, j

limiting distribution

We get the same limit no matter what the starting value is. Thus

limiting distribution (revised)

Equivalently, we say that a distribution π* is a limiting (aka equilibrium) distribution if

Limiting distribution (alternative definition)

where P is the transition matrix, no matter what the initial distribution π is. Also

Another alternative definition

This means that for any set of initial states

limit P^n

is a square matrix of identical rows, where each row is π*

Key Properties

Property 1: A limiting distribution must be a stationary distribution.

Proof:

Proof 1

Proof 2

Property 2: If a finite Markov chain is irreducible and aperiodic, then a unique stationary distribution  exists where πi = 1/ni, and this distribution is a limiting distribution.

 

Under construction

Links

↑ Markov chains

References

Aldridge, M. (2021) Long-term behaviour of Markov chains. Introduction to Markov Processes
https://mpaldridge.github.io/math2750/S11-long-term-chains.html

Norris, J. (2004) Discrete-time Markov chains. Cambridge University Press
https://www.statslab.cam.ac.uk/~jrn10//Markov/

Pishro-Nik, H. (2021) Stationary and limiting distributions
https://www.probabilitycourse.com/chapter11/11_2_6_stationary_and_limiting_distributions.php

Leave a Comment