- Kolmogorov’s criterion
In

probability theory ,**Kolmogorov's criterion**, named afterAndrey Kolmogorov , is atheorem inMarkov process es 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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