# class structure

Let ${({X}_{n})}_{n\ge 1}$ 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 $j$:

$$i\to j\iff P({X}_{n}=j:{X}_{0}=i)>0\mathit{\hspace{1em}}\text{for some}\mathit{\hspace{1em}}n\ge 0$$ |

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 |
---|---|

Canonical name | ClassStructure |

Date of creation | 2013-03-22 14:18:21 |

Last modified on | 2013-03-22 14:18:21 |

Owner | CWoo (3771) |

Last modified by | CWoo (3771) |

Numerical id | 12 |

Author | CWoo (3771) |

Entry type | Definition |

Classification | msc 60J10 |

Related topic | MarkovChain |

Defines | communicating class |

Defines | irreducible chain |

Defines | closed class |

Defines | absorbing state |