Index translation of a Markov chain

Question

Let $(X_n)_{n\in \mathbb{N}}$ be a markov chain. Does there holds

$\mathbb{P}(X_n=i \mid X_k=j)=\mathbb{P}(X_{n+m}=i \mid X_{k+m}=j)$ for $n,m \in \mathbb{N}$ and $k \leq n$?

Answer 1

Markov chains that have this property are said to be time-homogeneous (or sometimes stationary), and it is common to define Markov Chains in a way that assumes them to be time-homogenous. That said, it is also possible (and perfectly sensible) to define them without this property, so it is important that you double-check your definition to see whether it includes this property.

It is probably true that the most common and most useful Markov chains are time-homogeneous.