In the mathematical study of stochastic processes, a Harris chain is a Markov chain where the chain returns to a particular part of the state space an unbounded number of times. Harris chains are regenerative processes and are named after Theodore Harris. The theory of Harris chains and Harris recurrence is useful for treating Markov chains on general (possibly uncountably infinite) state spaces.
Definition Let { X n } {\displaystyle \{X_{n}\}} be a Markov chain on a general state space Ω {\displaystyle \Omega } with stochastic kernel K {\displaystyle K} . The kernel represents a generalized one-step transition probability law, so that P ( X n + 1 ∈ C ∣ X n = x ) = K ( x , C ) {\displaystyle P(X_{n+1}\in C\mid X_{n}=x)=K(x,C)} for all states x {\displaystyle x} in Ω {\displaystyle \Omega } and all measurable sets C ⊆ Ω {\displaystyle C\subseteq \Omega } . The chain { X n } {\displaystyle \{X_{n}\}} is a Harris chain if there exists A ⊆ Ω , ε > 0 {\displaystyle A\subseteq \Omega ,\varepsilon >0} , and probability measure ρ {\displaystyle \rho } with ρ ( Ω ) = 1 {\displaystyle \rho (\Omega )=1} such that
… excerpt ends here. Continue reading the full article.
