Difference between revisions of "Markov chain, class of zero states of a"
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{{MSC|60J10|60J27}} | {{MSC|60J10|60J27}} | ||
[[Category:Markov processes]] | [[Category:Markov processes]] | ||
− | A set | + | A set $ K $ |
+ | of states of a homogeneous [[Markov chain|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 | + | for any $ i , j \in K $, |
− | + | $$ | |
+ | p _ {il} ( t) = \ | ||
+ | {\mathsf P} \{ \xi ( t) = l \mid \xi ( 0) = i \} = 0 | ||
+ | $$ | ||
− | for any | + | for any $ i \in K $, |
+ | $ l \in S \setminus K $, | ||
+ | $ t > 0 $, | ||
+ | and | ||
− | + | $$ \tag{* } | |
+ | {\mathsf E} \tau _ {ii} = \infty | ||
+ | $$ | ||
− | for any | + | 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 | 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. | 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 < | + | 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: | 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 | + | 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==== | ====References==== | ||
− | + | {| | |
− | + | |valign="top"|{{Ref|C}}|| K.L. Chung, "Markov chains with stationary transition probabilities" , Springer (1967) {{MR|0217872}} {{ZBL|0146.38401}} | |
− | + | |} | |
====Comments==== | ====Comments==== | ||
Line 46: | Line 93: | ||
====References==== | ====References==== | ||
− | + | {| | |
+ | |valign="top"|{{Ref|F}}|| W. Feller, [[Feller, "An introduction to probability theory and its applications"|"An introduction to probability theory and its applications"]], '''1–2''', Wiley (1966) | ||
+ | |- | ||
+ | |valign="top"|{{Ref|Fr}}|| D. Freedman, "Markov chains", Holden-Day (1975) {{MR|0686269}} {{MR|0681291}} {{MR|0556418}} {{MR|0428472}} {{MR|0292176}} {{MR|0237001}} {{MR|0211464}} {{MR|0164375}} {{MR|0158435}} {{MR|0152015}} {{ZBL|0501.60071}} {{ZBL|0501.60069}} {{ZBL|0426.60064}} {{ZBL|0325.60059}} {{ZBL|0322.60057}} {{ZBL|0212.49801}} {{ZBL|0129.30605}} | ||
+ | |- | ||
+ | |valign="top"|{{Ref|I}}|| M. Iosifescu, "Finite Markov processes and their applications", Wiley (1980) {{MR|0587116}} {{ZBL|0436.60001}} | ||
+ | |- | ||
+ | |valign="top"|{{Ref|KS}}|| J.G. Kemeny, J.L. Snell, "Finite Markov chains", v. Nostrand (1960) {{MR|1531032}} {{MR|0115196}} {{ZBL|0089.13704}} | ||
+ | |- | ||
+ | |valign="top"|{{Ref|KSK}}|| J.G. Kemeny, J.L. Snell, A.W. Knapp, "Denumerable Markov chains", Springer (1976) {{MR|0407981}} {{ZBL|0348.60090}} | ||
+ | |- | ||
+ | |valign="top"|{{Ref|Re}}|| D. Revuz, "Markov chains", North-Holland (1975) {{MR|0415773}} {{ZBL|0332.60045}} | ||
+ | |- | ||
+ | |valign="top"|{{Ref|Ro}}|| V.I. Romanovsky, "Discrete Markov chains", Wolters-Noordhoff (1970) (Translated from Russian) {{MR|0266312}} {{ZBL|0201.20002}} | ||
+ | |- | ||
+ | |valign="top"|{{Ref|S}}|| E. Seneta, "Non-negative matrices and Markov chains", Springer (1981) {{MR|2209438}} {{ZBL|0471.60001}} | ||
+ | |} |
Latest revision as of 07:59, 6 June 2020
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=21650