# #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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