For reversible chains $P,\widetilde P$ with stationary laws $\pi,\widetilde\pi$, assign to every directed $\widetilde P$-edge $(x,y)$ a path $\Gamma_{xy}$ of $P$-edges. If
$$
A=\max_{e=(u,v)}
\frac1{\pi(u)P(u,v)}
\sum_{\Gamma_{xy}\ni e}
\widetilde\pi(x)\widetilde P(x,y)|\Gamma_{xy}|
$$
and $\widetilde\pi(x)\leq a\pi(x)$, the <Canonical paths comparison theorem> gives $\gamma(P)\geq\gamma(\widetilde P)/(aA)$, with equivalent conventions absorbing $a$ into the congestion.

Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 215 / 3 / b / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/iii/paper-215/3/b/solution