Definition (Stationary)

Given a Stochastic Process $X_n,n\ge0$. $X_n$ is a time homogeneous (or time-invariant) Markov chain if it is a Markov chain, and there exists $\{p_{ij}:i,j\in S\}$ such that for any $i,j\in S$, $$P(X_{n+1}=j|X_n=i)=p_{ij}, \mbox{ for any } n\ge0$$ where $p_{ij}$ is the probability of transitioning into state $j$ when in state $i$ or if its conditional pmfs $p_{X_{i}|X_{i-1}}$ do not depend on time index $i$: $$P(X_i=b|X_{i-1}=a)=P(X_2=b|X_{1}=a)\ \forall i\ge2, \ \forall a,b\in\mathcal{X}$$