Kingman's formula

In probability theory, Kolmogorov's criterion, named after Andrey Kolmogorov, is a theorem in 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

$p_{j_1{j}_2}{p}_{j_2{j}_3}\cdots p_{j_{n-1}{j}_n}{p}_{j_n{j}_1}=p_{j_1{j}_n}{p}_{j_n{j}_{n-1}}\cdots p_{j_3{j}_2}{p}_{j_2{j}_1}$

for all finite sequences of states

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

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

Example

File:Kolmogorov criterion dtmc.svg

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

$q_{j_1{j}_2}{q}_{j_2{j}_3}\cdots q_{j_{n-1}{j}_n}{q}_{j_n{j}_1}=q_{j_1{j}_n}{q}_{j_n{j}_{n-1}}\cdots q_{j_3{j}_2}{q}_{j_2{j}_1}$

for all finite sequences of states

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