Absorbing state

of a Markov chain

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)