# Minimization of the asymptotic variance in MCMC

Suppose $$(X_n)_{n\in\mathbb N_0}$$ is a Markov chain generated by the Metropolis-Hastings algorithm. Assume $$(X_n)_{n\in\mathbb N_0}$$ is stationary and consider the ergodic averages $$A_n:=\frac1n\sum_{i=0}^{n-1}f(X_i)$$ for some a priori fixed square-integrable $$f$$. Assume our goal is to minimize the asymptotic variance $$\sigma^2:=\lim_{n\to\infty}n\operatorname{Var}[A_n].$$

Tierney's theorem (see, for example, [1, Theorem 4], [2, Theorem 2.7] and [3, Lemma 2]) gives us a guideline how the transition kernel $$(X_n)_{n\in\mathbb N_0}$$ influences $$\sigma^2$$ - either in terms of nonnegative operators (in [1] and [2]) or the inverse Laplacian (in [3]).

However, [1] and [2] give a condition under which the asymptotic variance of every $$f$$ would be minimized and the corresponding minimization problem is complicated, since it depends on this additional parameter. So, my question is: Is there any easier condition on the transition kernel, depending solely on our fixed $$f$$, which ensures that $$\sigma^2$$ is minimized? [3] might be such a condition, but it involves the computation of the inverse Laplacian (which has no easy to handle closed form; does it?).

• Which set of kernels do you consider for the minimisation? – Xi'an Oct 15 '19 at 11:31
• @Xi'an I've asked for the concrete application here: mathoverflow.net/q/343651/91890. – 0xbadf00d Oct 15 '19 at 13:04