Kolmogorov's criterion

(Redirected from 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

edit

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

edit

 

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, 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

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