3.2 Chapman Kolmogrov Equations
\(n -\) Step transition probabilities \begin {align*} P_{ij}^n \, & = \, P\left (X_{n + m} = j\, \big /\, X_m = i\right )\hspace {0.5cm} \forall \, n > 0\, , \, \, \, i\,,\, \geq 0\\ & = P\left (X_n = j\, \big /\, X_0 = i\right ). \end {align*}
Chapman Komogrov equations
For any positive integers \(n\) and \(m\) and states \(i\) and \(j\),
\[P_{ij}^{n + m} \, = \, \sum _{k = 0}^{\infty }P_{ik}^m\, P_{kj}^n\]
Proof. \(\begin {aligned}[t] P_{ij}^{n + m} \, & = \, P\left (X_{n + m} = j\, \big /\, X_0 = i\right ) = \sum _{k = 0}^{\infty } P\left (X_{n + m} = j\, , \, X_m = k\, \big /\, X_0 = i\right ). \end {aligned}\) □
Questions on this section
Stuck on something here? Ask below and it stays attached to this topic.