Solution
= Solution
A time-homogeneous <Markov chain> on a countable <state space> $S$ is a sequence of $S$-valued <random variable>[random variables] for which, for every positive-probability finite history,
$$
P(X_{n+1}=j\mid X_0=i_0,\ldots,X_n=i)
=P(X_{n+1}=j\mid X_n=i)=P_{ij}.
$$
Thus the conditional next-step law depends only on the current state, and the transition probabilities do not depend on time.