On the convergence of the Markov chain simulation method
The following results on ergodicity of Markov chains with general state spaces have been proved. Suppose that the Markov chain \(\{X_n\}\) with state space \(({\mathcal X},{\mathcal B})\) and transition function \(P(x,C)\) has an invariant probability measure \(\pi\), and there is a set \(A\in{\mathcal B}\), a probability measure \(\rho\) with \(\rho(A)=1\), a constant \(\varepsilon>0\) and an integer \(n_0\geq1\) such that \(\pi\{x:P_x(T(A)<\infty)>0\}=1\), and \(P^{n_0}(x,\cdot)\geq\varepsilon\rho(\cdot)\) for each \(x\in A\), where \(T(A)=\inf\{n>0: X_n\in A\}\). Then \[ \lim_{n\to\infty} \sup_{C\in {\mathcal B}} \Biggl|{1\over n} \sum^n_{j=1} P^j(x,C)- \pi(C)\Biggr|=0\qquad \pi\text{-a.s.}. \] Let \(f(x)\) be a measurable function such that \(\int|f(y)|\pi(dy)<\infty\). Then \[ P_x\Biggl(\lim_{n\to\infty} {1\over n} \sum^n_{j=1} f(X_j)= \int f(y)\pi(dy)\Biggr)=1\qquad \pi\text{-a.s.}, \] and \[ \lim_{n\to\infty} {1\over n} \sum^n_{j=1} E_x(f(X_j))= \int f(y)\pi(dy)\qquad \pi\text{-a.s.}. \] In addition, suppose that \[ \text{g.c.d.}\{m:\text{ there is an }\varepsilon_m>0\text{ such that } P^m(x,\cdot)\geq \varepsilon_m\rho(\cdot)\text{ for each } x\in A\}=1. \] Then there is a set \(D\in {\mathcal B}\) such that \(\pi(D)=1\), and \[ \lim_{n\to\infty} \sup_{C\in{\mathcal B}} |P^n(x,C)-\pi(C)|=0\qquad\text{for each } x\in D. \] The authors argue that, compared with the earlier results on the topic, these results are more suitable to meet with the needs of the Markov chain simulation method, the assumptions made above are easier to check in reality.
scientific article; zbMATH DE number 922131
scientific article