# Kolmogorov’s criterion

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Kolmogorov’s criterion

In probability theory, Kolmogorov's criterion, named after Andrey Kolmogorov, is a theorem in Markov processes which states that a stationary Markov chain with transition matrix "P" and state space "S" is reversible if and only if its transition probabilities satisfy

: $p_\left\{j_1 j_2\right\} p_\left\{j_2 j_3\right\} cdots p_\left\{j_\left\{n-1\right\} j_n\right\} p_\left\{j_n j_1\right\} = p_\left\{j_1 j_n\right\} p_\left\{j_n j_\left\{n-1 cdots p_\left\{j_3 j_2\right\} p_\left\{j_2 j_1\right\}$

for any finite sequence

: $j_1, j_2, ldots, j_n in S$

of states.

A proof can be found in F.P. Kelly Reversibility and Stochastic Networks (Wiley, Chichester, 1979) ISBN 0471276014 [http://www.statslab.cam.ac.uk/~frank/BOOKS/kelly_book.html] p. 22

ee also

* Kolmogorov’s generalized criterion

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