= Solution
The sequence is a <Markov chain> when the conditional law of its next state depends on the past only through the current state. Precisely, for each history of positive <probability> and every $j\in S$,
$$
\boxed{\Pr(X_{n+1}=j\mid X_n=i_n,\ldots,X_0=i_0)
=\Pr(X_{n+1}=j\mid X_n=i_n).}
$$
A time-homogeneous chain additionally has transition probabilities independent of $n$, written $P_{ij}=\Pr(X_{n+1}=j\mid X_n=i)$. The displayed single transition matrix in the subsequent parts specifies this time-homogeneous case.
Back to article page
