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Markov chain, class of zero states of a

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2010 Mathematics Subject Classification: Primary: 60J10 Secondary: 60J27 [MSN][ZBL]

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,

References

[C] K.L. Chung, "Markov chains with stationary transition probabilities" , Springer (1967) MR0217872 Zbl 0146.38401

Comments

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

References

[F] W. Feller, "An introduction to probability theory and its applications", 1–2, Wiley (1966)
[Fr] D. Freedman, "Markov chains", Holden-Day (1975) MR0686269 MR0681291 MR0556418 MR0428472 MR0292176 MR0237001 MR0211464 MR0164375 MR0158435 MR0152015 Zbl 0501.60071 Zbl 0501.60069 Zbl 0426.60064 Zbl 0325.60059 Zbl 0322.60057 Zbl 0212.49801 Zbl 0129.30605
[I] M. Iosifescu, "Finite Markov processes and their applications", Wiley (1980) MR0587116 Zbl 0436.60001
[KS] J.G. Kemeny, J.L. Snell, "Finite Markov chains", v. Nostrand (1960) MR1531032 MR0115196 Zbl 0089.13704
[KSK] J.G. Kemeny, J.L. Snell, A.W. Knapp, "Denumerable Markov chains", Springer (1976) MR0407981 Zbl 0348.60090
[Re] D. Revuz, "Markov chains", North-Holland (1975) MR0415773 Zbl 0332.60045
[Ro] V.I. Romanovsky, "Discrete Markov chains", Wolters-Noordhoff (1970) (Translated from Russian) MR0266312 Zbl 0201.20002
[S] E. Seneta, "Non-negative matrices and Markov chains", Springer (1981) MR2209438 Zbl 0471.60001
How to Cite This Entry:
Markov chain, class of zero states of a. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Markov_chain,_class_of_zero_states_of_a&oldid=26564
This article was adapted from an original article by A.M. Zubkov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article