Questions about stationarity of a Markov process I was wondering for a Markov process:


*

*Is it true that:

it is a stationary process, iff its transition probability
  function is invariant to
  time-translation (i.e. it is
  homogeneous) and its initial
  distribution is invariant to
  time-translation?


*When the state space is discrete, is
the limit of its transition
probability function related to
stationarity for the process? Is its stationary
distribution related to stationarity, besides both have "stationary" in their names?


Thanks and regards!
 A: *

*Stationary in stationary process means that the distribution of the process is invariant by time-shifts, that is, the distributions of $(X_n)_{n\ge0}$ and $(X_{n+1})_{n\ge0}$ coincide. Hence stationarity requires the distribution of $X_n$ to be the same for every $n\ge0$ and also the distribution of $(X_n,X_{n+1})$ to be the same for every $n\ge0$. For Markov processes this is enough to guarantee that the transition probabilities are independent on $n$ and that the distribution of $X_0$ is invariant by this transition kernel. Reciprocally, if these two properties hold, the Markov chain is stationary. 

*The first question of this item makes no sense. Regarding the second question, if the distribution of $X_0$ is stationary in the sense that it is invariant with respect to the transition kernel, and if the transition kernel is independent on $n$, then obviously the Markov process is stationary.
Recall that for a measure $\nu$ and a transition kernel $p$, $\nu p$ is the measure defined by
$$(\nu p)(x)=\displaystyle\sum_y\nu(y)p(y,x),
$$ 
for every state $x$. The measure $\nu$ is said to be invariant by the transition kernel $p$ if $\nu p=\nu$.
Let me suggest that you read this.
A: *

*Markov Processes are not always stationary. It is only stationary if $P(X_{n+1} = b| X_n = a) = P(X_2 = b| X_1 = a)$. In other words $p(x_{n+1}|x_n)$ does not depend on $n$.

