Chapman–Kolmogorov equation

The Chapman - Kolmogorov equation in probability theory, an equation for the transition probabilities in Markov chains or more generally with Markov processes.

Markov chains

The Chapman - Kolmogorov equation for Markov chains, the probability of arrival in the post steps, starting in a state, as the sum of the possible paths with the intermediate station dar. Formally, this means:

Let be a Markov chain with transition matrix and state space.

Then for all

The proof of the equation is usually done as follows:

Using the definition of matrix multiplication on the transition matrix results

Was being utilized in that the following applies to all of.

Markov processes

For a general Markov process with semigroup of transition kernels, the Chapman - Kolmogorov equation can also be written in short as

The composition of nuclei called. Induction can be derived from it, that