Chapman-Kolmogorov Equation

Theorem
Let $X$ be a discrete state-space Markov Chain with with $n$-step transition probability matrix:
 * $\mathbf P^{\paren n} = \sqbrk {\map {p^{\paren n} } {j, k} }_{j, k \mathop \in S}$

where:
 * $\map {p^{\paren n} } {j, k} = \mathbb P \sqbrk {X_{m + n} = k \mid X_m = j} = {p_{j k} }^{\paren n}$ is the $n$-step transition probability.

Then:
 * $\mathbf P^{\paren {n + m} } = \mathbf P^{\paren n} \mathbf P^{\paren m}$

or equivalently:
 * $\displaystyle {p_{i j} }^{\paren {n + m} } = \sum_{k \mathop \in S} {p_{i k} }^{\paren n} {p_{k j} }^{\paren m}$

Proof
We consider the conditional probability on the left hand side: