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# 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.

## Mathematics Subject Classification

60J10*no label found*

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