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
Stationary Markov chain Markov chain with transition probabilities that represent the "long run" fraction in state
The stationary fraction in state is given by the solution uniqueness.
Consider a Markov chain whose stationary distribution is given by
Stationary Markov chain =
In plain english:
If we have a Markov chain whose stationary distribution exists, then the stationary Markov chain has the solutions written in every entry of it's column.
Say in the above example we had sample space where lower class, middle class upper class. Then the stationary Markov chain represents the fraction of people who will be in the lower, upper and middle classes as in the long run of the population will be in the lower class.
Thank you both for your answers.
Actually what Focus said is what im interested in.
We have an irreducible, recurrent and stationary Markov process with finite space state S and invariant distribution on . We consider to be the canonical space given by Kolmogorov's Extension theorem for .
My question about stationarity comes after introducing the concept of reversibility:
is reversible if has the same distribution as for all , and .
A measurable transformation is introduced (time shift) :
Then it is stated:
Since is stationary, is -invariant, i.e. .
My question: What role does stationarity play so it makes our - invariant and how does this follows from the definition of stationarity?? That is why i first asked about the formal definition of stationarity.