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I was wondering if equilibrium distribution, steady-state distribution, stationary distribution and limiting distribution mean the same, or there are differences between?

I learn them in the context of Discrete-time Markov Chain as far as I know. Or they also appear in other situations of stochastic processes and probability?

In the Wikipedia page for Markov Chain, there seems not very clear to me in defining and using these concepts.


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So far as I know, a stationary distribution is invariant under the Markov chain evolution, i.e., $\bar{\pi} = P \bar{\pi}$, but not every stationary distribution is necessarily the distribution the chain converges to in the limit (limiting distribution). There are conditions under which both are the same. – RandomGuy Feb 22 '11 at 16:35
@RandomGuy: Thanks! (1) if both exist, under what conditions will the two be the same, and in what cases will they not? (2) Which one's existence implies the existence of the other, or under what conditions this will be true? – Tim Apr 20 '11 at 2:11
up vote 6 down vote accepted

(1) You forgot one! In the index to Gregory Lawler's book Introduction to Stochastic Processes (2nd edition) we find

  • equilibrium distribution, see invariant distribution
  • stationary distribution, see invariant distribution
  • steady state distribution, see invariant distribution

All this terminology is for one concept; a probability distribution that satisfies $\pi=\pi P$. In other words, if you choose the initial state of the Markov chain with distribution $\pi$, then the process is stationary. I mean if $X_0$ is given distribution $\pi$, then $X_n$ has distribution $\pi$ for all $n\geq 0$. Such a $\pi$ exists if and only if the chain has a positive recurrent state. An invariant distribution need not be unique. For example, if the Markov chain has $n<\infty$ states, the collection $\{\pi: \pi=\pi P\ \}$ is a non-empty simplex in $\mathbb{R}^n$ whose extreme points (corners) correspond to recurrent classes.

(2) The concept of a limiting distribution is related, but not exactly the same. Suppose that $\pi_j:=\lim_n P_{ij}^n$ exists and doesn't depend on $i$. These are called limiting probabilities and the vector $\pi:=(\pi_1,\dots,\pi_n)$ will satisfy $\pi=\pi P$. So a limiting distribution (if it exists) is always invariant. Limiting probabilities exist when the chain is irreducible, positive recurrent, and aperiodic.

A typical case when the limiting distribution fails to exist is when the chain is periodic. For instance, for the two state chain with transition matrix $P=\pmatrix{0&1\cr 1&0}$ the unique invariant distribution is $\pi=(1/2,1/2)$, but $P_{ij}^n$ alternates between $0$ and $1$ so fails to converge.

I'm not sure that all authors use these terms in the same way, so you want to be careful when reading other books.

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Hi, I know this is a lot later than when you posted this but I have a question. Does this post mean that in order to calculate the limiting distribution and stationary distributions, we use same procedure of working out the $\pi = \pi P$, but in certain cases this is called limiting distribution and certain times it is the stationary distribution? – Kaish Jan 22 '13 at 21:30
@Kaish: I think you're right. Every limiting distribution is invariant (or stationary), but not every invariant distribution can be obtained from iterating the transition matrix (i.e., not all of them are limiting distributions). An invariant or stationary distribution is defined by $\pi = \pi P$, so they'll always satisfy that. – cd98 Nov 10 '15 at 21:18

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