# Absorbing state

of a Markov chain $\xi (t)$

2010 Mathematics Subject Classification: Primary: 60J10 [MSN][ZBL]

A state $i$ such that

$${\mathsf P} \{ \xi (t) = i \mid \xi (s) = i \} = 1 \ \textrm{ for } \textrm{ any } t \geq s.$$

An example of a Markov chain with absorbing state $0$ 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 $S$ of states of a homogeneous Markov chain $\xi (t)$ with discrete time and transition probabilities

$$p _ {ij} = {\mathsf P} \{ \xi (t+1) = j \mid \xi (t) = i \} ,$$

in which a subset $H$ is distinguished and suppose one has to find the probabilities

$$q _ {ih} = {\mathsf P} \{ \xi ( \tau (H)) = h \mid \xi (0) = i \} , \ i \in S,\ h \in H,$$

where $\tau (H) = \mathop{\rm min} \{ {t > 0 } : {\tau (t) \in H } \}$ is the moment of first hitting the set $H$. If one introduces the auxiliary Markov chain $\xi ^ {*} (t)$ differing from $\xi (t)$ only in that all states $h \in H$ are absorbing in $\xi ^ {*} (t)$, then for $h \in H$ the probabilities

$$p _ {ih} ^ {*} (t) = {\mathsf P} \{ \xi ^ {*} (t) = h \mid \xi ^ {*} (0) = i \} =$$

$$= \ {\mathsf P} \{ \tau (H) \leq t, \xi ( \tau (H)) = h \mid \xi (0) = i \}$$

are monotonically non-decreasing for $t \uparrow \infty$ and

$$\tag{* } q _ {ih} = \lim\limits _ {t \rightarrow \infty } p _ {ih} ^ {*} (t), \ i \in S,\ h \in H.$$

By virtue of the basic definition of a Markov chain

$$p _ {ih} ^ {*} (t + 1) = \ \sum _ {j \in S } p _ {ij} p _ {ih} ^ {*} (t), \ t \geq 0,\ i \in S \setminus H,\ h \in H,$$

$$p _ {hh} ^ {*} (t) = 1,\ h \in H; \ p _ {ih} ^ {*} (t) = 0,\ i, h \in H, i \neq h.$$

The passage to the limit for $t \rightarrow \infty$ taking into account (*) gives a system of linear equations for $q _ {ih}$:

$$q _ {ih} = \sum _ {j \in S } p _ {ij} q _ {ih} ,\ \ i \in S \setminus H,\ h \in H,$$

$$q _ {hh} = 1,\ h \in H ; \ q _ {ih} = 0,\ i, h \in H, i \neq h.$$

#### References

 [F] W. Feller, "An introduction to probability theory and its applications", 1, Wiley (1968)
How to Cite This Entry:
Absorbing state. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Absorbing_state&oldid=45007
This article was adapted from an original article by A.M. Zubkov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article