Difference between revisions of "Absorbing state"
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− | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> | + | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> W. Feller, "An introduction to probability theory and its applications" , '''1''' , Wiley (1968) {{MR|0228020}} {{ZBL|0155.23101}} </TD></TR></table> |
Revision as of 10:29, 27 March 2012
of a Markov chain
2020 Mathematics Subject Classification: Primary: 60J10 [MSN][ZBL]
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) MR0228020 Zbl 0155.23101 |
Absorbing state. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Absorbing_state&oldid=23573