Solution
= Solution
A time-homogeneous <Markov chain> on a countable state space satisfies, whenever the conditioning event has positive probability,
$$
P(X_{n+1}=j\mid X_0=i_0,\ldots,X_n=i)=P(X_{n+1}=j\mid X_n=i)=p_{ij},
$$
where the last probabilities do not depend on $n$. Thus the present state contains all the past information relevant to the next transition. Time homogeneity is the convention used here, since the problem specifies a single <transition matrix>.