Bounty: 50
Let $(E,mathcal E,mu)$ be a probability space, $(X_n){ninmathbb N_0}$ be an $(E,mathcal E)$-valued ergodic time-homogeneous Markov chain with stationary distribution $mu$ and $$A_nf:=frac1nsum{i=0}^{n-1}f(X_i);;;text{for }finmathcal L^1(mu)text{ and }ninmathbb N.$$ Suppose we’re estimating an integral $$mu f=int f:{rm d}mu$$ for some $finmathcal L^1(mu)$ and that $$sqrt n(A_nf-mu f)xrightarrow{ntoinfty}mathcal N(0,sigma^2(f))tag1$$ with $$sigma^2(f)=lim_{ntoinfty}noperatorname{Var}[A_nf].$$
Assume that the support $B:={fne0}$ is “small”, but $mu(B)>0$. Now let $$f_B:=1_Bleft(f-frac{mu(1_Bf)}{mu(B)}right).$$ Instead of $sigma^2(f)$ we could consider $sigma^2(f_B)$ which, by definition, should tell us something about the deviation of $f=1_Bf$ from its mean.
If our concern is the minimization of the asymptotic variance of our estimation of $mu f$, does it make sense to consider $sigma^2(f_B)$ instead of $sigma^2(f)$? How are $sigma^2(f_B)$ and $sigma^2(f)$ related?
Background:
My actual concern is that I need to estimate $lambda(hf)$ for a variety of $mathcal E$-measurable $h:Eto[0,infty)$, where $lambda$ is a reference measure on $(E,mathcal E)$ so that $mu$ has a density $p$ with respect to $lambda$ and ${p=0}subseteq{f=0}$. Now the support $E_0:={p>0}cap{h>0}$ is very “small” and you may assume that $h=1_{{:h:>:0:}}$. I want to estimate $lambda(hf)$ using the importance sampling estimator.
Say $(X_n){ninmathbb N_0}$ is the chain generated by the Metropolis-Hastings algorithm with target distribution $mu$, $(Y_n){ninmathbb N}$ is the corresponding proposal sequence and $Z_n:=(X_{n-1},Y_n)$ for $ninmathbb N$. If the proposal kernel is denoted by $Q$, we know that $(Z_n)_{ninmathbb N}$ has stationary distribution $nu:=muotimes Q$.
Now, in light of the special form of the to be estimated integral $lambda(hf)$, I thought it might be tempting to consider the Markov chain corresponding to the successive times of its returns to the set $Etimes E_0$. To be precise, let $tau_0:=0$ and $$tau_k:=infleft{n>tau_{k-1}:Y_nin E_0right};;;text{for }kinmathbb N.$$ Assuming that $Etimes E_0$ is a recurrent set for $(Z_n)_{ninmathbb N}$, we know that $left(Z_{tau_k}right)_{kinmathbb N}$ is again an ergodic time-homogeneous Markov chain with stationary distribution $$nu_0:=frac{left.nuright|_{Etimes E_0}}{nu(Etimes E_0)}.$$
I wondered whether it makes sense to build an estimator using $left(Z_{tau_k}right){kinmathbb N}$ instead of $(Z_n){ninmathbb N}$ or if this would gain nothing.
Get this bounty!!!
