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[[Category:Markov chains]]

A [[Markov chain|Markov chain]] in which a random trajectory $\xi(t)$, starting at any state $\xi(0)=i$, returns to that state with probability 1. In terms of the [[Transition probabilities|transition probabilities]] $p_{ij}(t)$, recurrence of a discrete-time Markov chain is equivalent to the divergence for any $i$ of the series

$$\sum_{t=0}^\infty p_{ij}(t).$$

In a recurrent Markov chain a trajectory $\xi(t)$, $0\leq t<\infty$, $\xi(0)=i$, returns infinitely often to the state $i$ 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 $x$ to $x\pm1$ with probabilities $1/2$; in the symmetric walk on the plane a particle moves from $(x,y)$ to one of the four points $(x\pm1,y)$, $(x,y\pm1)$ with probabilities $1/4$. 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 $(x,y,z)$ to a neighbouring point $(x\pm1,y,z)$, $(x,y\pm1,z)$, $(x,y,z\pm1)$ is equal to $1/6$, is not recurrent. In this case the probability of return of the particle to its initial point is approximately 0.35.

====References====
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|valign="top"|{{Ref|F}}|| W. Feller, [[Feller, "An introduction to probability theory and its  applications"|"An introduction to probability theory and its  applications"]], '''1''' , Wiley (1966)
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====References====
{|
|valign="top"|{{Ref|Fr}}|| D. Freeman, "Markov chains" , Holden-Day (1975)
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|valign="top"|{{Ref|I}}|| M. Iosifescu, "Finite Markov processes and their applications" , Wiley (1980) {{MR|0587116}} {{ZBL|0436.60001}}
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|valign="top"|{{Ref|KS}}|| J.G. Kemeny, J.L. Snell, "Finite Markov chains" , v. Nostrand (1960) {{MR|1531032}} {{MR|0115196}} {{ZBL|0089.13704}}
|-
|valign="top"|{{Ref|KSK}}|| J.G. Kemeny, J.L. Snell, A.W. Knapp, "Denumerable Markov chains" , Springer (1976) {{MR|0407981}} {{ZBL|0348.60090}}
|-
|valign="top"|{{Ref|Re}}|| D. Revuz, "Markov chains" , North-Holland (1975) {{MR|0415773}} {{ZBL|0332.60045}}
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|valign="top"|{{Ref|Ro}}|| V.I. Romanovsky, "Discrete Markov chains" , Wolters-Noordhoff (1970) (Translated from Russian) {{MR|0266312}} {{ZBL|0201.20002}}
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|valign="top"|{{Ref|Se}}|| E. Seneta, "Non-negative matrices and Markov chains" , Springer (1981) {{MR|2209438}} {{ZBL|0471.60001}}
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|valign="top"|{{Ref|Sp}}|| V. Spitzer, "Principles of random walk" , v. Nostrand (1964) {{MR|0171290}} {{ZBL|0119.34304}}
|}