  We assume a drift condition towards a small set and bound the mean square
error of estimators obtained by taking averages along a single trajectory of a
Markov chain Monte Carlo algorithm. We use these bounds to construct
fixed-width nonasymptotic confidence intervals. For a possibly unbounded
function $f:\stany \to R,$ let $I=\int_{\stany} f(x) \pi(x) dx$ be the value of
interest and $\hat{I}_{t,n}=(1/n)\sum_{i=t}^{t+n-1}f(X_i)$ its MCMC estimate.
Precisely, we derive lower bounds for the length of the trajectory $n$ and
burn-in time $t$ which ensure that $$P(|\hat{I}_{t,n}-I|\leq \varepsilon)\geq
1-\alpha.$$ The bounds depend only and explicitly on drift parameters, on the
$V-$norm of $f,$ where $V$ is the drift function and on precision and
confidence parameters $\varepsilon, \alpha.$ Next we analyse an MCMC estimator
based on the median of multiple shorter runs that allows for sharper bounds for
the required total simulation cost. In particular the methodology can be
applied for computing Bayesian estimators in practically relevant models. We
illustrate our bounds numerically in a simple example.
