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

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The theorem states that an irreducible, positive recurrent, aperiodic Markov chain with transition matrix P is reversible if and only if its stationary Markov chain satisfies[1]

 

for all finite sequences of states

 

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

That is, the chain-multiplication along any cycle is the same forwards and backwards.

Example

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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,

 

Proof

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Let   be the Markov chain and denote by   its stationary distribution (such exists since the chain is positive recurrent).

If the chain is reversible, the equality follows from the relation  .

Now assume that the equality is fulfilled. Fix states   and  . Then

      .

Now sum both sides of the last equality for all possible ordered choices of   states  . Thus we obtain   so  . Send   to   on the left side of the last. From the properties of the chain follows that  , hence   which shows that the chain is reversible.

Continuous-time Markov chains

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The theorem states that a continuous-time Markov chain with transition rate matrix Q is, under any invariant probability vector, reversible if and only if its transition probabilities satisfy[1]

 

for all finite sequences of states

 

The proof for continuous-time Markov chains follows in the same way as the proof for discrete-time Markov chains.

References

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