\begin{definition}[Definition:Markov Chain]
Let $\sequence {X_n}_{n \mathop \ge 0}$ be a stochastic process over a countable set $S$.
Let $\map \Pr X$ denote the probability of the random variable $X$.
Let $\sequence {X_n}_{n \mathop \ge 0}$ satisfy the Markov property:
:$\condprob {X_{n + 1} = i_{n + 1} } {X_0 = i_0, X_1 = i_1, \ldots, X_n = i_n} = \condprob {X_{n + 1} = i_{n + 1} } {X_n = i_n}$
for all $n \ge 0$ and all $i_0, i_1, \ldots, i_{n + 1} \in S$.
That is, such that the conditional probability of $X_{i + 1}$ is dependent only upon $X_i$ and upon no earlier values of $\sequence {X_n}$.
That is, the state of $\sequence {X_n}$ in the future is unaffected by its history.
Then $\sequence {X_n}_{n \mathop \ge 0}$ is a '''Markov chain'''.
\end{definition}