Before revision: The Markov-Chain Monte-Carlo (MCMC) method has been used widely in the literature for various applications, in particular estimating the expectation \mathbb{E_{\pi}[f] of a } {\rm  )}(}{\cal  {\cal   function f: \Omega \to [a,b] over a distribution \pi on \Omega (a. k.a . mean-estimation), to within \varepsilon additive error (w.h.p.). Letting R \doteq b-a, standard variance-agnostic MCMC mean-estimators run the chain for (\frac{TR^{2}{\varepsilon^{2}}) steps, when given as inputan (often loose) upper-bound T on the relaxation time \tau}  _{  rel } . When an upper-bound V on the stationary variance v_{\pi \doteq \mathbb{V}_{\pi}[f] is known, } \frac{TR{\varepsilon}+\frac{TV}{\varepsilon^{2}}} steps suffice.
After revision: The Markov-Chain Monte-Carlo (MCMC) method has been used widely in the literature for various applications, in particular estimating the expectation \mathbb{E_{\pi}[f] of a } {\rm )}(}{\cal {\cal function f: \Omega \to [a,b] over a distribution \pi on \Omega (a. k.a . The inter-trace variance can be efficiently estimated from observed data and leads to a more efficient MCMC-mean estimator. Prior MCMC mean-estimators run the chain for (\frac{TR^{2}{\varepsilon^{2}}) steps, when given as inputan (often loose) upper-bound T on the relaxation time \tau} _{  rel } . When an upper-bound V on the stationary variance v_{\pi \doteq \mathbb{V}_{\pi}[f] is known, } \frac{TR{\varepsilon}+\frac{TV}{\varepsilon^{2}}} steps suffice.