Difference between revisions of "Absorbing state"
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''of a Markov chain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010430/a0104301.png" />'' | ''of a Markov chain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010430/a0104301.png" />'' | ||
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A state <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010430/a0104302.png" /> such that | A state <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010430/a0104302.png" /> such that |
Revision as of 07:09, 7 January 2012
of a Markov chain
[ 2010 Mathematics Subject Classification MSN: 60J10 | MSCwiki: 60J10 ]
A state such that
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An example of a Markov chain with absorbing state is a branching process.
The introduction of additional absorbing states is a convenient technique that enables one to examine the properties of trajectories of a Markov chain that are associated with hitting some set.
Example. Consider the set of states of a homogeneous Markov chain
with discrete time and transition probabilities
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in which a subset is distinguished and suppose one has to find the probabilities
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where is the moment of first hitting the set
. If one introduces the auxiliary Markov chain
differing from
only in that all states
are absorbing in
, then for
the probabilities
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are monotonically non-decreasing for and
![]() | (*) |
By virtue of the basic definition of a Markov chain
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The passage to the limit for taking into account (*) gives a system of linear equations for
:
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References
[1] | W. Feller, "An introduction to probability theory and its applications" , 1 , Wiley (1968) |
Absorbing state. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Absorbing_state&oldid=20022