# Definition:Absorbing State

## Definition

Let $\sequence {X_n}_{n \mathop \ge 0}$ be a Markov chain on a state space $S$.

Let $i \in S$ be an element of the state space $S$.

Then $i$ is an absorbing state of $\sequence {X_n}$ if and only if:

$X_k = i \implies X_{k + 1} = i$

That is, it is an element of $S$ such that if $\sequence {X_n}$ reaches $i$, it stays there.

## Also known as

An absorbing state can also be seen referred to as an absorbing barrier.

## Also defined as

Some sources define an absorbing state on a random walk only.

Some sources separate the definitions of absorbing state and absorbing barrier, using the former for a Markov chain and the latter for a random walk.