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I'm running the Metropolis-Hastings algorithm with state space $E$, target distribution $\mu=p\lambda$ and proposal kernel $Q$ to estimate $\mu(hf)$ for a fixed function $f:E\to[0,\infty)^3$ and a finite number of bounded $h:E\to[0,\infty)$. $p$ and the proposal kernel $Q$ both depend on a parameter which I want to choose such that the asymptotic variance of the estimates is reduced as much as possible or even minimized.

How should I tackle this problem? Which estimator $A_ng$ for $\mu g$ should I use?

Let $\sigma^2(g):=\lim_{n\to\infty}n\operatorname{Var}[A_ng]$. Once I've fixed an estimator, I'm willing to assume that $\sigma^2(|f|)<\infty$ ($|\;\cdot\;|$ denoting the Euclidean norm). This should yield that $\sigma^2(hf_i)<\infty$ for all bounded $h:E\to[0,\infty)$ and $i=1,2,3$ and hence that a CLT holds for all estimates of interest.

I know several stronger conditions ensuring that a CLT holds for a broad class of functions; see, for example, section 5.2 in General state space Markov chains and MCMC algorithms by Gareth O. Roberts and Jeffrey S. Rosenthal. Moreover, a useful criterion on asymptotic variance reduction is given in Theorem 4 of A note on Metropolis-Hastings kernels for general state spaces by Luke Tierney.

However, these approaches leave me with the problem of minimizing the supremum of an integral functional. For example, if we use the ordinary ergodic average $A_ng:=\frac1n\sum_{i=0}^{n-1}g(X_i)$, $(X_n)_{n\in\mathbb N_0}$ being the Metropolis-Hastings chain, then Theorem 25 in the paper of Roberts and Rosenthal yields that we could try to minimize $$\sup_{g\in L^2(\mu)\setminus\{0\}}\frac{\left\|\kappa g\right\|_{L^2(\mu)}}{\left\|g\right\|_{L^2(\mu)}}\tag1.$$ The supremum occuring in $(1)$ makes it hard to choose the parameter minimizing this expression.

Since minimizing $(1)$ would yield way more than I need (it would yield minimum worst-case asymptotic variance for all $g\in L^2(\mu)$, but I'm only interested in estimates for $hf_i$), I hope that there is a better approach.

So, what I would need is to find some nice expression for the worst-case asymptotic variance of $A_n(hf_i)$, for all bounded $h:E\to[0,\infty)$ and $i=1,2,3$. Maybe a nice upper bound only involving $f$.

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