Kolmogorov’s criterion


Kolmogorov’s criterion

In probability theory, Kolmogorov's criterion, named after Andrey Kolmogorov, is a theorem in Markov processes which states that a stationary Markov chain with transition matrix "P" and state space "S" 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 any finite sequence

: j_1, j_2, ldots, j_n in S

of states.

A proof can be found in F.P. Kelly Reversibility and Stochastic Networks (Wiley, Chichester, 1979) ISBN 0471276014 [http://www.statslab.cam.ac.uk/~frank/BOOKS/kelly_book.html] p. 22

ee also

* Kolmogorov’s generalized criterion


Wikimedia Foundation. 2010.

Look at other dictionaries:

  • 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… …   Wikipedia

  • Andrey Kolmogorov — Infobox Scientist name = Andrey Kolmogorov birth date = birth date|1903|4|25 birth place = Tambov, Imperial Russia nationality = Russian death date = death date and age|1987|10|20|1903|4|25 death place = Moscow, USSR field = Mathematician work… …   Wikipedia

  • Cramér–von Mises criterion — In statistics the Cramér–von Mises criterion is a criterion used for judging the goodness of fit of a cumulative distribution function F * compared to a given empirical distribution function Fn, or for comparing two empirical distributions. It is …   Wikipedia

  • Cramér-von-Mises criterion — In statistics the Cramér von Mises criterion is a form of minimum distance estimation used for judging the goodness of fit of a probability distribution F^* compared to a given distribution F is given by:n W^2 = n int { infty}^{infty} [F(x)… …   Wikipedia

  • Minimum distance estimation — (MDE) is a statistical method for fitting a mathematical model to data, usually the empirical distribution. Contents 1 Definition 2 Statistics used in estimation 2.1 Chi square criterion …   Wikipedia

  • List of mathematics articles (K) — NOTOC K K approximation of k hitting set K ary tree K core K edge connected graph K equivalence K factor error K finite K function K homology K means algorithm K medoids K minimum spanning tree K Poincaré algebra K Poincaré group K set (geometry) …   Wikipedia

  • Occam's razor — For the aerial theatre company, see Ockham s Razor Theatre Company. It is possible to describe the other planets in the solar system as revolving around the Earth, but that explanation is unnecessarily complex compared to the modern consensus… …   Wikipedia

  • List of statistics topics — Please add any Wikipedia articles related to statistics that are not already on this list.The Related changes link in the margin of this page (below search) leads to a list of the most recent changes to the articles listed below. To see the most… …   Wikipedia

  • Boris Chirikov — Boris Valerianovich Chirikov (Russian Борис Валерианович Чириков)(June 6, 1928 ndash; February 12, 2008) was an outstanding Soviet and Russian physicist. He was the founder of the physical theory of Hamiltonian chaos and made pioneering… …   Wikipedia

  • Group method of data handling — (GMDH) is a family of inductive algorithms for computer based mathematical modeling of multi parametric datasets that features fully automatic structural and parametric optimization of models. GMDH is used in such fields as data mining, knowledge …   Wikipedia