Can someone please provide a reference to a detailed proof of the CLT for MC? I have a paper MC's for MCMC'ists but I have a problem understanding a crucial part. Namely, the transition from line 2 to line 3 here
So I am looking for a different source where this transition will be better explained. Does anybody know of a source with a detailed proof? I do not want a source, littered with "it can be shown" and "it follows from (something not in the text)...", but something I can learn from. The attached source is great, but I cannot understand the mentioned transition.
The step that interests you is rather direct. To see why it holds, first rewrite the double sum in the expectation as $$S=\sum_{k=0}^\infty\sum_{m=1}^\infty f(X_k) f(X_{k+m})\mathbf 1_{T>k+m}$$ then condition each $k$-term by $\sigma(X_k)$. This yields $$E(S)=\sum_{k=0}^\infty\sum_{m=1}^\infty E\left(f(X_k) E\left( f(X_{k+m})\mathbf 1_{T>k+m}\mid X_k\right)\right)$$ By the Markov property and the homogeneity of the Markov chain, for each $k$ and each positive $m$, $$E\left( f(X_{k+m})\mathbf 1_{T>k+m}\mid X_k\right)=\mathbf 1_{T>k}g_m(X_k)$$ where $$g_m(x)=E\left(f(X_m)\mathbf 1_{T>m}\mid X_0=x\right)$$ This is the formula in your text, minus the typo $Y_0=0$.