Difference between revisions of "Transition with prohibitions"
Ulf Rehmann (talk | contribs) m (MR/ZBL numbers added) |
(MSC|60J10|60J35 Category:Markov chains) |
||
Line 1: | Line 1: | ||
''transition with taboo states, for a Markov chain'' | ''transition with taboo states, for a Markov chain'' | ||
+ | |||
+ | {{MSC|60J10|60J35}} | ||
+ | |||
+ | [[Category:Markov chains]] | ||
The set of trajectories of the [[Markov chain|Markov chain]] that never enters in a specified set of states in a given time interval. Let, for example, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093790/t0937901.png" /> be a Markov chain with discrete time and set of states <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093790/t0937902.png" />, while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093790/t0937903.png" /> is the set of "taboo" states (the taboo set). Then the taboo probabilities <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093790/t0937904.png" /> are | The set of trajectories of the [[Markov chain|Markov chain]] that never enters in a specified set of states in a given time interval. Let, for example, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093790/t0937901.png" /> be a Markov chain with discrete time and set of states <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093790/t0937902.png" />, while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093790/t0937903.png" /> is the set of "taboo" states (the taboo set). Then the taboo probabilities <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093790/t0937904.png" /> are |
Revision as of 15:22, 17 April 2012
transition with taboo states, for a Markov chain
2020 Mathematics Subject Classification: Primary: 60J10 Secondary: 60J35 [MSN][ZBL]
The set of trajectories of the Markov chain that never enters in a specified set of states in a given time interval. Let, for example, be a Markov chain with discrete time and set of states
, while
is the set of "taboo" states (the taboo set). Then the taboo probabilities
are
![]() |
![]() |
The properties of the taboo probabilities are analogous to those of the ordinary transition probabilities
, since the families of matrices
and
,
, form multiplication semi-groups; however, while
,
. Different problems, e.g. the study of the distribution of the time to the first entrance of the Markov chain into a given set or limit theorems for branching processes (cf. Branching process) under conditions of non-extinction, in fact amount to the investigation of various properties of taboo probabilities.
References
[1] | K.L. Chung, "Markov chains with stationary transition probabilities" , Springer (1960) MR0116388 Zbl 0092.34304 |
Comments
References
[a1] | I.I. [I.I. Gikhman] Gihman, A.V. [A.V. Skorokhod] Skorohod, "The theory of stochastic processes" , 1 , Springer (1975) (Translated from Russian) MR0375463 Zbl 0305.60027 |
Transition with prohibitions. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Transition_with_prohibitions&oldid=23671