Kolmogorov's criterion

In probability theory, Kolmogorov's criterion, named after Andrey Kolmogorov, is a theorem in Markov processes concerning stationary Markov chains (i.e. ones with stationary transition probabilities, also called "time-homogeneous Markov chains"). The theorem states that such a chain 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 all finite sequences of states

$j_1, j_2, \ldots, j_n \in S .$

Here p_ij are elements of the transition matrix P and S is the state space of the chain.

Proofs of this theorem are available in the literature.^[1]

Notes