Markov chain, recurrent

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A Markov chain in which a random trajectory , starting at any state , returns to that state with probability 1. In terms of the transition probabilities , recurrence of a discrete-time Markov chain is equivalent to the divergence for any of the series

In a recurrent Markov chain a trajectory , , , returns infinitely often to the state with probability 1. In a recurrent Markov chain there are no inessential states and the essential states decompose into recurrent classes. An example of a recurrent Markov chain is the symmetric random walk on the integer lattice on the line or plane. In the symmetric walk on the line a particle moves from position to with probabilities ; in the symmetric walk on the plane a particle moves from to one of the four points , with probabilities . In these examples a particle, starting the walk at an arbitrary point, returns to that point with probability 1. The symmetric walk on the integer lattice in the three-dimensional space, when the probability of transition from to a neighbouring point , , is equal to , is not recurrent. In this case the probability of return of the particle to its initial point is approximately 0.35.


[1] W. Feller, "An introduction to probability theory and its applications" , 1 , Wiley (1966)



[a1] D. Freeman, "Markov chains" , Holden-Day (1975)
[a2] M. Iosifescu, "Finite Markov processes and their applications" , Wiley (1980)
[a3] J.G. Kemeny, J.L. Snell, "Finite Markov chains" , v. Nostrand (1960)
[a4] J.G. Kemeny, J.L. Snell, A.W. Knapp, "Denumerable Markov chains" , Springer (1976)
[a5] D. Revuz, "Markov chains" , North-Holland (1975)
[a6] V.I. [V.I. Romanovskii] Romanovsky, "Discrete Markov chains" , Wolters-Noordhoff (1970) (Translated from Russian)
[a7] E. Seneta, "Non-negative matrices and Markov chains" , Springer (1981)
[a8] V. Spitzer, "Principles of random walk" , v. Nostrand (1964)
How to Cite This Entry:
Markov chain, recurrent. Encyclopedia of Mathematics. URL:,_recurrent&oldid=11703
This article was adapted from an original article by B.A. Sevast'yanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article