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