# Thread: Definition of Stationary markov chain

1. ## Definition of Stationary markov chain

hello

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!

2. Originally Posted by mabruka
hello

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 $\displaystyle \mathbb{P}(X_n | X_{n-1}=x)=\mathbb{P}(X_{n-1}|X_{n-2}=x)$.

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

3. Originally Posted by mabruka
hello

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 $\displaystyle =$ Markov chain with transition probabilities that represent the "long run" fraction in state $\displaystyle i.$

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

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

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

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

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

Nahwatimsayin?

Actually what Focus said is what im interested in.

Context:

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

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

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

A measurable transformation is introduced (time shift) :

$\displaystyle \theta_t:\Omega\longrightarrow \Omega$

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

Then it is stated:

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

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

5. Originally Posted by mabruka

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 $\displaystyle \mathbb{P}(X_n | X_{n-1}=x)=\mathbb{P}(X_{n-1}|X_{n-2}=x).$

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

6. So stationarity and time homogeneiety are two different definitions, but

stationarity implies time homogeneiety right?