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Difference between revisions of "Transition with prohibitions"

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''transition with taboo states, for a Markov chain''
 
''transition with taboo states, for a Markov chain''
  
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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
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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
  
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093790/t0937905.png" /></td> </tr></table>
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093790/t0937905.png" /></td> </tr></table>
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====References====
 
====References====
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<table><TR><TD valign="top">[1]</TD> <TD valign="top"> K.L. Chung,   "Markov chains with stationary transition probabilities" , Springer (1960)</TD></TR></table>
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<table><TR><TD valign="top">[1]</TD> <TD valign="top"> K.L. Chung, "Markov chains with stationary transition probabilities" , Springer (1960) {{MR|0116388}} {{ZBL|0092.34304}} </TD></TR></table>
  
  
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====References====
 
====References====
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<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> I.I. [I.I. Gikhman] Gihman,   A.V. [A.V. Skorokhod] Skorohod,   "The theory of stochastic processes" , '''1''' , Springer (1975) (Translated from Russian)</TD></TR></table>
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<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> I.I. [I.I. Gikhman] Gihman, A.V. [A.V. Skorokhod] Skorohod, "The theory of stochastic processes" , '''1''' , Springer (1975) (Translated from Russian) {{MR|0375463}} {{ZBL|0305.60027}} </TD></TR></table>

Revision as of 10:32, 27 March 2012

transition with taboo states, for a Markov chain

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
How to Cite This Entry:
Transition with prohibitions. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Transition_with_prohibitions&oldid=23671
This article was adapted from an original article by A.M. Zubkov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article
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