Kolmogorov's criterion

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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[edit]

The theorem states that a irreducible, positive recurrent, aperiodic Markov chain with transition matrix P is reversible if and only if its transition probabilities satisfy[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.


Example[edit]

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,

Proof[edit]

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[edit]

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]

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[edit]

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