Kolmogorov’s generalized criterion

Kolmogorov’s generalized criterion

Kolmogorov's generalized criterion is a proposition in Markov processes which states that a stationary Markov process with state space S and generator matrix Q has reversed process with generator matrix Q' if and only if

# scriptstyle{q'_{i} = q_{i for every state scriptstyle{i in S} (where scriptstyle{q_{i} = -q_{ii).
# For every finite sequence of states scriptstyle{i_{1}, i_{2}, ldots, i_{n} in S}, the condition scriptstyle{q_{i_1 i_2} q_{i_2 i_3} cdots q_{i_{n-1} i_n} q_{i_n i_1} = q'_{i_1 i_n} q'_{i_n i_{n-1 cdots q'_{i_3 i_2} q'_{i_2 i_1 holds.

References

A proof can be found in P.G. Harrison. "Turning Back Time in Markovian Process Algebra," Theoretical Computer Science, 290(3): 1947-1986, January 2003. [http://pubs.doc.ic.ac.uk/rcat/rcat.pdf]

ee also

* Kolmogorov’s criteria


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