Markov chain, class of zero states of a

2020 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

[1]	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

[a1]	W. Feller, "An introduction to probability theory and its applications", 1–2, Wiley (1966)
[a2]	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
[a3]	M. Iosifescu, "Finite Markov processes and their applications", Wiley (1980) MR0587116 Zbl 0436.60001
[a4]	J.G. Kemeny, J.L. Snell, "Finite Markov chains", v. Nostrand (1960) MR1531032 MR0115196 Zbl 0089.13704
[a5]	J.G. Kemeny, J.L. Snell, A.W. Knapp, "Denumerable Markov chains", Springer (1976) MR0407981 Zbl 0348.60090
[a6]	D. Revuz, "Markov chains", North-Holland (1975) MR0415773 Zbl 0332.60045
[a7]	V.I. [V.I. Romanovskii] Romanovsky, "Discrete Markov chains", Wolters-Noordhoff (1970) (Translated from Russian) MR0266312 Zbl 0201.20002
[a8]	E. Seneta, "Non-negative matrices and Markov chains", Springer (1981) MR2209438 Zbl 0471.60001