Markov chain, class of zero states of a

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A set of states of a homogeneous Markov chain with state space such that

for any ,

for any , , , and


for any , where is the return time to the state :

for a discrete-time Markov chain, and

for a continuous-time Markov chain.

As in the case of a class of positive states (in the definition of a positive class (*) is replaced by ), states belonging to the same zero class have a number of common properties. For example, for any states of a zero class ,

An example of a Markov chain whose states form a single zero class is the symmetric random walk on the integers:

where are independent random variables,


[1] K.L. Chung, "Markov chains with stationary transition probabilities" , Springer (1967)


Cf. also Markov chain, class of positive states of a.


[a1] W. Feller, "An introduction to probability theory and its applications" , 1–2 , Wiley (1966)
[a2] D. Freedman, "Markov chains" , Holden-Day (1975)
[a3] M. Iosifescu, "Finite Markov processes and their applications" , Wiley (1980)
[a4] J.G. Kemeny, J.L. Snell, "Finite Markov chains" , v. Nostrand (1960)
[a5] J.G. Kemeny, J.L. Snell, A.W. Knapp, "Denumerable Markov chains" , Springer (1976)
[a6] D. Revuz, "Markov chains" , North-Holland (1975)
[a7] V.I. [V.I. Romanovskii] Romanovsky, "Discrete Markov chains" , Wolters-Noordhoff (1970) (Translated from Russian)
[a8] E. Seneta, "Non-negative matrices and Markov chains" , Springer (1981)
How to Cite This Entry:
Markov chain, class of zero states of a. Encyclopedia of Mathematics. URL:,_class_of_zero_states_of_a&oldid=18811
This article was adapted from an original article by A.M. Zubkov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article