Monte Carlo Methods in Statistics

is converging to the expectation Ef [h(X)] when T goes to infinity. Furthermore, the precision of this approximation is exactly of the same kind as the precision of a statistical estimate, in that it usually evolves as O( √ T ). Therefore, once a sample x1, . . . , xT is produced according to a distribution density f , all standard statistical tools, including bootstrap, apply to this sample (with the further appeal that more data points can be produced if deemed necessary). As illustrated by Figure 1, the variability due to a single Monte Carlo experiment must be accounted for, when drawing conclusions about its output and evaluations