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
 |
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
 |
in which a subset 
is distinguished and suppose one has to find the probabilities
 |
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
 |
 |
are monotonically non-decreasing for 
and
 | (*) |
By virtue of the basic definition of a Markov chain
 |
 |
The passage to the limit for 
taking into account (*) gives a system of linear equations for 
:
 |
 |
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