#StackBounty: #variance #monte-carlo #asymptotics #central-limit-theorem How are the variance of an estimate to $int_Bf:{rm }dmu$ a…

Bounty: 50

Let $(E,mathcal E,mu)$ be a probability space, $(X_n){ninmathbb N_0}$ be an $(E,mathcal E)$-valued stationary time-homogeneous Markov chain with initial distribution 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?

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