Kolmogorov's criterion

In probability theory, Kolmogorov's criterion, named after Andrey Kolmogorov, is a theorem giving a necessary and sufficient condition for a Markov chain or continuous-time Markov chain to be stochastically identical to its time-reversed version.

Discrete-time Markov chains

The theorem states that a Markov chain with transition matrix P is reversible if and only if its transition probabilities satisfy^[1]

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

for all finite sequences of states

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

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

Example

Consider this figure depicting a section of a Markov chain with states i, j, k and l and the corresponding transition probabilities. Here Kolmogorov's criterion implies that the product of probabilities when traversing through any closed loop must be equal, so the product around the loop i to j to l to k returning to i must be equal to the loop the other way round,

$p_{ij}p_{jl}p_{lk}p_{ki} = p_{ik}p_{kl}p_{lj}p_{ji}.$

Continuous-time Markov chains

The theorem states that a continuous-time Markov chain with transition rate matrix Q is reversible if and only if its transition probabilities satisfy^[1]

$q_{j_1 j_2} q_{j_2 j_3} \cdots q_{j_{n-1} j_n} q_{j_n j_1} = q_{j_1 j_n} q_{j_n j_{n-1}} \cdots q_{j_3 j_2} q_{j_2 j_1}$

for all finite sequences of states

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

References

1. Kelly, Frank P. (1979). Reversibility and Stochastic Networks (PDF). Wiley, Chichester. pp. 21–25.