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Difference between revisions of "Absorbing state"

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====References====
 
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<table><TR><TD valign="top">[1]</TD> <TD valign="top"> W. Feller,   "An introduction to probability theory and its applications" , '''1''' , Wiley (1968)</TD></TR></table>
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<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 :

References

[1] W. Feller, "An introduction to probability theory and its applications" , 1 , Wiley (1968) MR0228020 Zbl 0155.23101
How to Cite This Entry:
Absorbing state. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Absorbing_state&oldid=23573
This article was adapted from an original article by A.M. Zubkov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article