# Markov chain, recurrent

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.

#### References

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

#### Comments

#### References

[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: http://encyclopediaofmath.org/index.php?title=Markov_chain,_recurrent&oldid=11703