Chapman–Kolmogorov equation

Statement

Let $(X_n)_{n \ge 0}$ be a time-homogeneous Markov chain on a countable state space $S$ . For $n \ge 0$ , let $p^{(n)}_{ij} = P(X_{m+n} = j \mid X_m = i)$ be the $n$ -step transition probabilities. Then for all $m, n \ge 0$ and $i, j \in S$ , $p^{(m+n)}_{ij} = \sum_{k \in S} p^{(m)}_{ik} \, p^{(n)}_{kj}.$ In matrix form, $P^{(m+n)} = P^{(m)} P^{(n)}$ , so $P^{(n)} = P^n$ for the transition matrix $P$ .

Remarks

Proof: condition on the state $X_m = k$ at the intermediate time, with the law of total probability, and use the Markov property. The equation says that to go from $i$ to $j$ in $m + n$ steps, the chain must be at some state $k$ after $m$ steps. For a Markov process with transition kernels $p_{s,t}(x, dy)$ , the equation reads $p_{s,u}(x, A) = \int p_{s,t}(x, dy) \, p_{t,u}(y, A)$ for $s \le t \le u$ .

History

The equation is named after Sydney Chapman and Andrey Kolmogorov.

Tag	`01APZ8GDQH`
URI	`m2://01APZ8GDQH`
Aliases	`chapman-kolmogorov-equations`
Kind	statement, theorem
Writing	W1 stated
MSC	60J10
Status	This copy of the site has no build data. Run `m2 status 01APZ8GDQH`.
Uses	Markov chain, Transition matrix, Law of total probability, Markov property
Same as	Wikidata Q722882 Wikipedia Chapman–Kolmogorov equation

Tag

01APZ8GDQH

URI

m2://01APZ8GDQH

Aliases

chapman-kolmogorov-equations

Kind

statement, theorem

Writing

W1 stated

MSC

60J10

Status

This copy of the site has no build data. Run m2 status 01APZ8GDQH.

Uses

Markov chain, Transition matrix, Law of total probability, Markov property

Same as

Wikidata Q722882

Wikipedia Chapman–Kolmogorov equation

Chapman–Kolmogorov equation

Statement

Remarks

History

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