The worst-case distance to stationarity is
$$
d(t)=\max_\sigma\|P^t(\sigma,\cdot)-\pi_\beta\|_{\rm TV}.
$$
The <Path coupling theorem> says that if a coupling contracts an integer-valued path metric by a factor $\alpha<1$ for every adjacent pair, then
$$
d(t)\leq\operatorname{diam}(\Omega)\alpha^t.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 215 / 2 / b / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/iii/paper-215/2/b/solution