Markov chain, class of zero states of a
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 |
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