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
 |
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 
:
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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