Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 216 / 3 / a / Solution

Markov chain Monte Carlo asymptotic variance for a stationary Markov chain and $\psi \in L^2(\pi )$ iswhenever the limit and series exist. If a reversible Markov chain has positive $L_0^2(\pi )$ spectral gap $\gamma$, then the spectral theorem for normal operators on a separable Hilbert space gives