Chapman-Kolmogorov Equation

Theorem
Let $X$ be a discrete state-space Markov Chain with with $n$-step transition probability matrix $\mathbf{P}^{\left(n\right)}=\left[p^{\left(n\right)}\left(j,k\right)\right]_{j,k\in S}$, where $p^{\left(n\right)}\left(j,k\right)=\mathbb{P}\left[X_{m+n}=k\middle|X_m=j\right]=p_{jk}^{\left(n\right)}$ is the $n$-step transition probability.

Then $\mathbf{P}^{\left(n+m\right)}=\mathbf{P}^{\left(n\right)}\mathbf{P}^{\left(m\right)}$, or equivalently $\displaystyle p_{ij}^{\left(n+m\right)}=\sum_{k\in S}p_{ik}^{\left(n\right)}p_{kj}^{\left(m\right)}$.

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