Markov chain, class of zero states of a
2020 Mathematics Subject Classification: Primary: 60J10 Secondary: 60J27 [MSN][ZBL]
A set $ K $ of states of a homogeneous Markov chain $ \xi ( t) $ with state space $ S $ such that
$$ {\mathsf P} \{ {\exists t > 0 } : {\xi ( t) = j \mid \xi ( 0) = i } \} = 1 $$
for any $ i , j \in K $,
$$ p _ {il} ( t) = \ {\mathsf P} \{ \xi ( t) = l \mid \xi ( 0) = i \} = 0 $$
for any $ i \in K $, $ l \in S \setminus K $, $ t > 0 $, and
$$ \tag{* } {\mathsf E} \tau _ {ii} = \infty $$
for any $ i \in K $, where $ \tau _ {ii} $ is the return time to the state $ i $:
$$ \tau _ {ii} = \min \ \{ {t > 0 } : {\xi ( t) = i \mid \xi ( 0) = i } \} $$
for a discrete-time Markov chain, and
$$ \tau _ {ii} = \inf \ \{ {t > 0 } : {\xi ( t) = i \mid \xi ( 0) = i , \xi ( 0 + ) \neq i } \} $$
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 $ {\mathsf E} \tau _ {ii} < \infty $), states belonging to the same zero class have a number of common properties. For example, for any states $ i , j $ of a zero class $ K $,
$$ \lim\limits _ {t \rightarrow \infty } p _ {ij} ( t) = 0 . $$
An example of a Markov chain whose states form a single zero class is the symmetric random walk on the integers:
$$ \xi ( 0) = 0 ,\ \ \xi ( t) = \xi ( t - 1 ) + \eta ( t) ,\ \ t = 1 , 2 \dots $$
where $ \eta ( 1) , \eta ( 2) \dots $ are independent random variables,
$$ {\mathsf P} \{ \eta ( i) = 1 \} \ = {\mathsf P} \{ \eta ( i) = - 1 \} \ = 1/2 ,\ i = 1 , 2 ,\dots . $$
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://encyclopediaofmath.org/index.php?title=Markov_chain,_class_of_zero_states_of_a&oldid=47766