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.