Chapman-Kolmogorov Equation
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Theorem
Let $X$ be a homogeneous Markov chain with $n$-step transition probability matrix:
- $\mathbf P^{\paren n} = \sqbrk { {p_{j k} }^{\paren n} }_{j, k \mathop \in S}$
where:
- ${p_{j k} }^{\paren n} = \condprob {X_n = k} {X_0 = j} $ is the $n$-step transition probability.
Then:
- $\mathbf P^{\paren {n + m} } = \mathbf P^{\paren n} \mathbf P^{\paren m}$
or equivalently:
- $\ds {p_{i j} }^{\paren {n + m} } = \sum_{k \mathop \in S} {p_{i k} }^{\paren n} {p_{k j} }^{\paren m}$
Proof
We consider the conditional probability on the left hand side:
\(\ds \ds {p_{i j} }^{\paren {n + m} }\) | \(=\) | \(\ds \map \Pr {X_{m + n} = j \mid X_0 = i}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \condprob {\paren {\bigcup_{k \mathop \in S} \sqbrk {X_{n + m} = j, X_n = k} } } {X_0 = i}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{k \mathop \in S} \condprob {X_{n + m} = j, X_n = k} {X_0 = i}\) | Definition of Countable Additivity | |||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{k \mathop \in S} \condprob {X_{n + m} = j} {X_n = k, X_0 = i} \times \condprob {X_n = k} {X_0 = i}\) | Chain Rule for Probability | |||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{k \mathop \in S} \condprob {X_{n + m} = j} {X_n = k} \times \condprob {X_n = k} {X_0 = i}\) | Markov Property | |||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{k \mathop \in S} \condprob {X_{0 + m} = j} {X_0 = k} \times \condprob {X_n = k} {X_0 = i}\) | Transition Probabilities of Homogeneous Markov Chain | |||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{k \mathop \in S} {p_{i k} }^{\paren n} {p_{k j} }^{\paren m}\) |
$\blacksquare$
Source of Name
This entry was named for Sydney Chapman and Andrey Nikolaevich Kolmogorov.