(Note that nothing yet has been assumed about the temporal (or any other) ordering of the random variables—the above equation applies equally to the marginalization of any of them.)
Application to time-dilated Markov chains
When the stochastic process under consideration is Markovian, the Chapman–Kolmogorov equation is equivalent to an identity on transition densities. In the Markov chain setting, one assumes that i1 < ... < in. Then, because of the Markov property,
where the conditional probability is the transition probability between the times . So, the Chapman–Kolmogorov equation takes the form
Informally, this says that the probability of going from state 1 to state 3 can be found from the probabilities of going from 1 to an intermediate state 2 and then from 2 to 3, by adding up over all the possible intermediate states 2.
When the probability distribution on the state space of a Markov chain is discrete and the Markov chain is homogeneous, the Chapman–Kolmogorov equations can be expressed in terms of (possibly infinite-dimensional) matrix multiplication, thus:
where P(t) is the transition matrix of jump t, i.e., P(t) is the matrix such that entry (i,j) contains the probability of the chain moving from state i to state j in t steps.
As a corollary, it follows that to calculate the transition matrix of jump t, it is sufficient to raise the transition matrix of jump one to the power of t, that is
The differential form of the Chapman–Kolmogorov equation is known as a master equation.
Pavliotis, Grigorios A. (2014). "Markov Processes and the Chapman–Kolmogorov Equation". Stochastic Processes and Applications. New York: Springer. pp. 33–38. ISBN978-1-4939-1322-0.
Ross, Sheldon M. (2014). "Chapter 4.2: Chapman−Kolmogorov Equations". Introduction to Probability Models (11th ed.). Academic Press. p. 187. ISBN978-0-12-407948-9.