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''transition with taboo states, for a Markov chain''
 
''transition with taboo states, for a Markov chain''
  
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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{{MSC|60J10|60J35}}
  
<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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[[Category:Markov chains]]
  
<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/t0937906.png" /></td> </tr></table>
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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,  $  \xi ( t) $
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be a Markov chain with discrete time and set of states  $  S $,
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while  $  H $
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is the set of "taboo" states (the taboo set). Then the taboo probabilities  $  {} _ {H} p _ {ij} ( t) $
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are
  
The properties of the taboo probabilities <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093790/t0937907.png" /> are analogous to those of the ordinary [[Transition probabilities|transition probabilities]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093790/t0937908.png" />, since the families of matrices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093790/t0937909.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093790/t09379010.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093790/t09379011.png" />, form multiplication semi-groups; however, while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093790/t09379012.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093790/t09379013.png" />. 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|Branching process]]) under conditions of non-extinction, in fact amount to the investigation of various properties of taboo probabilities.
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$$
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{} _ {H} p _ {ij} ( t= {\mathsf P} \{ \xi ( k) \notin
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H  ( k = 1 \dots t- 1 ),\
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\xi ( t) = j \mid  \xi ( 0) = i \} ,
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$$
  
====References====
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$$
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> K.L. Chung, "Markov chains with stationary transition probabilities" , Springer (1960) {{MR|0116388}} {{ZBL|0092.34304}} </TD></TR></table>
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i, j  \in  S.
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$$
  
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The properties of the taboo probabilities  $  {} _ {H} p _ {ij} ( t) $
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are analogous to those of the ordinary [[Transition probabilities|transition probabilities]]  $  p _ {ij} ( t) $,
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since the families of matrices  $  P( t) = \| p _ {ij} ( t) \| _ {i,j \in S }  $
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and  $  P _ {H} ( t) = \| {} _ {H} p _ {ij} ( t) \| _ {i,j \in S\setminus  H }  $,
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$  t \geq  0 $,
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form multiplication semi-groups; however, while  $  \sum _ {j \in S }  p _ {ij} ( t) = 1 $,
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$  \sum _ {j \in S }  {} _ {H} p _ {ij} ( t) \leq  1 $.
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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|Branching process]]) under conditions of non-extinction, in fact amount to the investigation of various properties of taboo probabilities.
  
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====References====
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{|
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|valign="top"|{{Ref|C}}|| K.L. Chung, "Markov chains with stationary transition probabilities" , Springer (1960) {{MR|0116388}} {{ZBL|0092.34304}}
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|}
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
<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>
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{|
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|valign="top"|{{Ref|GS}}|| I.I. Gihman, A.V. Skorohod, "The theory of stochastic processes" , '''1''' , Springer (1975) (Translated from Russian) {{MR|0375463}} {{ZBL|0305.60027}}
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|}

Latest revision as of 08:26, 6 June 2020


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, $ \xi ( t) $ be a Markov chain with discrete time and set of states $ S $, while $ H $ is the set of "taboo" states (the taboo set). Then the taboo probabilities $ {} _ {H} p _ {ij} ( t) $ are

$$ {} _ {H} p _ {ij} ( t) = {\mathsf P} \{ \xi ( k) \notin H ( k = 1 \dots t- 1 ),\ \xi ( t) = j \mid \xi ( 0) = i \} , $$

$$ i, j \in S. $$

The properties of the taboo probabilities $ {} _ {H} p _ {ij} ( t) $ are analogous to those of the ordinary transition probabilities $ p _ {ij} ( t) $, since the families of matrices $ P( t) = \| p _ {ij} ( t) \| _ {i,j \in S } $ and $ P _ {H} ( t) = \| {} _ {H} p _ {ij} ( t) \| _ {i,j \in S\setminus H } $, $ t \geq 0 $, form multiplication semi-groups; however, while $ \sum _ {j \in S } p _ {ij} ( t) = 1 $, $ \sum _ {j \in S } {} _ {H} p _ {ij} ( t) \leq 1 $. 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

[C] K.L. Chung, "Markov chains with stationary transition probabilities" , Springer (1960) MR0116388 Zbl 0092.34304

Comments

References

[GS] I.I. Gihman, A.V. 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