Kolmogorov’s criterion

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_{j_1 j_2} p_{j_2 j_3} cdots p_{j_{n-1} j_n} p_{j_n j_1} = p_{j_1 j_n} p_{j_n j_{n-1 cdots p_{j_3 j_2} p_{j_2 j_1}

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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