The summand intended in the question is $d,h(X(t))/S(X(t))$. Under stationarity,
$$
\mathbb E_\pi\left[\frac{d,h(X)}{S(X)}\right]
=\int\frac{d,h(x)}{S(x)}\frac{S(x)}d\nu(dx)
=\int h\,d\nu=H.
$$
Moreover $S(x)\geq dc_1$, so the summand is bounded by $1/c_1$. A stationary geometrically ergodic Markov chain satisfies the <Markov-chain law of large numbers>; hence
$$
\widehat H_n=\frac1n\sum_{t=1}^n
\frac{d,h(X(t))}{S(X(t))}
\longrightarrow H
$$
almost surely, and therefore in probability. In particular,
$$
\boxed{\Pr(|\widehat H_n-H|>\epsilon)\to0.}
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 216 / 3 / b / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2019/iii/paper-216/3/b/solution