# class structure

Let $(X_{n})_{n\geq 1}$ be a stationary Markov chain  and let $i$ and $j$ be states in the indexing set. We say that $i$ leads to $j$ or $j$ is accessible  from $i$, and write $i\to j$, if it is possible for the chain to get from state $i$ to state $j$:

 $i\to j\iff P(X_{n}=j:X_{0}=i)>0\quad\textrm{for some}\quad n\geq 0$

If $i\to j$ and $j\to i$ we say $i$ communicates with $j$ and write $i\leftrightarrow j$. $\leftrightarrow$ is an equivalence relation  (easy to prove). The equivalence classes  of this relation  are the communicating classes of the chain. If there is just one class, we say the chain is an irreducible chain.

A class $C$ is a closed class if $i\in C$ and $i\to j$ implies that $j\in C$ “Once the chain enters a closed class, it cannot leave it”

A state $i$ is an absorbing state if $\{i\}$ is a closed class.

Title class structure ClassStructure 2013-03-22 14:18:21 2013-03-22 14:18:21 CWoo (3771) CWoo (3771) 12 CWoo (3771) Definition msc 60J10 MarkovChain communicating class irreducible chain closed class absorbing state