Definition:Markov Chain/Homogeneous

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Let $\sequence {X_n}_{n \mathop \ge 0}$ be a Markov chain on a countable set $S$.

$\sequence {X_n}_{n \mathop \ge 0}$ is homogeneous if and only if $\condprob {X_{n + 1} = j} {X_n = i}$ does not depend on the value of $n$ for all $i, j \in S$.

That is: the transition probabilities are constant.