Chapman–Kolmogorov equation

In mathematics, specifically in the theory of Markovian in probability theory, the Chapman–Kolmogorov equation is an identity relating the joint probability distributions of different sets of coordinates on a stochastic process. The equation was derived independently by both the British mathematician Sydney Chapman and the Russian mathematician Andrey Kolmogorov.

Suppose that { f_i } is an indexed collection of random variables, that is, a stochastic process. Let

be the joint probability density function of the values of the random variables f₁ to f_n. Then, the Chapman–Kolmogorov equation is

i.e. a straightforward marginalization over the nuisance variable.

(Note that we have not yet assumed anything about the temporal (or any other) ordering of the random variables — the above equation applies equally to the marginalization of any of them.)

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 < ... < i_n. Then, because of the Markov property,

where the conditional probability ${\displaystyle p_{i;j}(f_{i}\mid f_{j})}$ is the transition probability between the times ${\displaystyle i>j}$. So, the Chapman–Kolmogorov equation takes the form

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