<Markov chain Monte Carlo asymptotic variance> for a stationary <Markov chain> and $\psi\in L^2(\pi)$ is
$$
\sigma_K^2(\psi)
=\lim_{n\to\infty}n\operatorname{Var}\left(
\frac1n\sum_{j=1}^n\psi(X_j)\right)
=\operatorname{Var}_\pi(\psi)
+2\sum_{k=1}^{\infty}
\operatorname{Cov}_\pi(\psi(X_0),\psi(X_k)),
$$
whenever the limit and series exist. If a <reversible Markov chain> has positive $L^2_0(\pi)$ <spectral gap> $\gamma$, then the <spectral theorem for normal operators on a separable Hilbert space> gives
$$
\sigma_K^2(\psi)
\leq\left(\frac2\gamma-1\right)
\operatorname{Var}_\pi(\psi).
$$

Solution

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

{c}
{title2=$\sigma_K^2(\psi)$}

For a stationary chain, the asymptotic variance of the sample mean is $\operatorname{Var}_\pi\psi+2\sum_{k\geq1}\operatorname{Cov}_\pi(\psi(X_0),\psi(X_k))$ when the series converges.


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