Harris chain

A Harris chain , named after the mathematician Theodore E. Harris , is a special Markov chain in discrete time on a measurable state space. Harris chains are interesting because you can formulate ergodic sentences for them.

definition

Be a measurable space . Let be a Markov chain on the state space with a transition kernel . Then the Harris chain is called if there are sets , a and a probability measure on with such that: ${\displaystyle (S,\Sigma )}$ ${\displaystyle (X_{n})_{n\in \mathbb {N} _{0}}}$ ${\displaystyle (S,\Sigma )}$ ${\displaystyle P}$ ${\displaystyle (X_{n})_{n\in \mathbb {N} _{0}}}$ ${\displaystyle A,B\in \Sigma }$ ${\displaystyle \varepsilon >0}$ ${\displaystyle \rho }$ ${\displaystyle (S,\Sigma )}$ ${\displaystyle \rho (B)=1}$

1. And applies to everyone ${\displaystyle x_{0}\in S}$ ${\displaystyle {\text{P}}(\tau _{A}<\infty \mid X_{0}=x_{0})>0}$
2. for all and all measurable applies ${\displaystyle x\in A}$ ${\displaystyle C\subseteq B}$ ${\displaystyle P(x,C)\geq \varepsilon \rho (C)\,.}$

In this case referred to the first time of entry of the chain in the amount . ${\displaystyle \tau _{A}=\inf\{n\in \mathbb {N} _{0}:X_{n}\in A\}}$ ${\displaystyle A}$

Individual evidence

1. ↑ Rick Durret: Probability: Theory and Examples. 4th edition. Cambridge University Press, 2010, ISBN 978-0-521-76539-8 , Section 6.8, p. 318ff ( limited preview in Google Book Search).