Before revision:  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:  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} receive, as input, upper-bounds on \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.