  I consider the problem of integrating a function $f$ over the $d$-dimensional
unit cube. I describe a multilevel Monte Carlo method that estimates the
integral with variance at most $\epsilon^{2}$ in
$O(d+\ln(d)d_{t}\epsilon^{-2})$ time, for $\epsilon>0$, where $d_{t}$ is the
truncation dimension of $f$. In contrast, the standard Monte Carlo method
typically achieves such variance in $O(d\epsilon^{-2})$ time. A lower bound of
order $d+d_{t}\epsilon^{-2}$ is described for a class of multilevel Monte Carlo
methods.
