Definition of Stationary markov chain

**mabruka** · April 9th 2010, 11:37 AM

hello

im a little confused about this definition.

Is a staionary markov chain a chain with a stationary distribution (not necessarily unique)?

or the stationary distribution needs to be unique (so the chain is irreducible and positive recurrent) to be a stationary markov chain?

or is something totally different?

Ive read something about being P-invariant under "time translations" but i can not find this definition formally.

any help would be great

thank you!

**Focus** · April 9th 2010, 12:24 PM

Originally Posted by mabruka

hello

im a little confused about this definition.

Is a staionary markov chain a chain with a stationary distribution (not necessarily unique)?

or the stationary distribution needs to be unique (so the chain is irreducible and positive recurrent) to be a stationary markov chain?

or is something totally different?

Ive read something about being P-invariant under "time translations" but i can not find this definition formally.

any help would be great

thank you!

From my knowledge, stationary refers to time homogeneity which is $\mathbb{P}(X_n | X_{n-1}=x)=\mathbb{P}(X_{n-1}|X_{n-2}=x)$ .

A context may help in determining if this is what you are after.

**Anonymous1** · April 9th 2010, 01:22 PM

Originally Posted by mabruka

hello

im a little confused about this definition.

Is a staionary markov chain a chain with a stationary distribution (not necessarily unique)?

or the stationary distribution needs to be unique (so the chain is irreducible and positive recurrent) to be a stationary markov chain?

or is something totally different?

Ive read something about being P-invariant under "time translations" but i can not find this definition formally.

any help would be great

thank you!

In general:
Stationary Markov chain $=$ Markov chain with transition probabilities that represent the "long run" fraction in state $i.$

The stationary fraction in state $i$ is given by the solution $\pi (i) \Rightarrow$ uniqueness.

For definiteness:
Consider a Markov chain whose stationary distribution is given by $\vec\pi = [1/4 \ \ 2/4 \ \ 1/4]<br />$

$\Rightarrow$ Stationary Markov chain = $\lim_{n\rightarrow \infty} p^n (i, j)= \left( \begin{array}{ccc} <br /> 1/4 & 2/4 & 1/4 \\ 1/4 & 2/4 & 1/4 \\ 1/4 & 2/4 & 1/4 \end{array} \right)$

In plain english:
If we have a Markov chain whose stationary distribution exists, then the stationary Markov chain has the solutions $\pi (i)$ written in every entry of it's $i^{th}$ column.

Say in the above example we had sample space $S=\{1 \ \ 2 \ \ 3\}$ where $1 =$ lower class, $2 =$ middle class $3 =$ upper class. Then the stationary Markov chain represents the fraction of people who will be in the lower, upper and middle classes as $t\rightarrow \infty.$ $e.g.$ in the long run $1/4$ of the population will be in the lower class.

Nahwatimsayin?

**mabruka** · April 9th 2010, 04:25 PM

Thank you both for your answers.

Actually what Focus said is what im interested in.

Context:

We have $\{\xi_t\}_{t\in \mathbb R}$ an irreducible, recurrent and stationary Markov process with finite space state S and invariant distribution $\Pi=\{\pi_i\}_{i\in S}$ on $(\Omega,\mathbb F, \mathbb P)$ . We consider $\Omega$ to be the canonical space given by Kolmogorov's Extension theorem for $\xi$ .

My question about stationarity comes after introducing the concept of reversibility:

$\xi$ is reversible if $(\xi_{t_1},\ldots,\xi_{t_k})$ has the same distribution as $(\xi_{T-t_1},\ldots,\xi_{T-t_k})$ for all $k\geq1$ , $t_1<\ldots,<t_k$ and $T\in \mathbb R$ .

A measurable transformation is introduced (time shift) :

$\theta_t:\Omega\longrightarrow \Omega$

$\theta_t \omega(s) =\omega(s+t)$

Then it is stated:

Since $\xi$ is stationary, $\mathbb P$ is $\theta_t$ -invariant, i.e. $\theta_t \mathbb P =\mathbb P$ .

My question: What role does stationarity play so it makes our $\mathbb P$ $\theta_t$ - invariant and how does this follows from the definition of stationarity?? That is why i first asked about the formal definition of stationarity.

**Anonymous1** · April 9th 2010, 06:31 PM

Originally Posted by mabruka

Thank you both for your answers.

Actually what Focus said is what im interested in.

Okay...

I assume you realize that the definition of stationary Markov chain that I gave directly implies time homogeneity, or $\mathbb{P}(X_n | X_{n-1}=x)=\mathbb{P}(X_{n-1}|X_{n-2}=x).$

I was simply trying to paint a vivid picture...

**mabruka** · April 11th 2010, 04:48 PM

So stationarity and time homogeneiety are two different definitions, but

stationarity implies time homogeneiety right?

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