Non-asymptotic Error Estimates for Monte Carlo Methods.

Summary In order to effectively implement Monte Carlo methods, the random approximation errors must be controlled. For this purpose, theoretical results are provided for the estimation of the number of simulations necessary to obtain a desired accuracy with a prescribed confidence interval. Therefore absolute, i.e., non-asymptotic, versions of the Central Limit Theorem (CLT) are developed: Berry–Esseen’s and Bikelis’ theorems, as well as concentration inequalities obtained from logarithmic Sobolev inequalities. The difficult subject of variance reduction techniques for Monte Carlo methods arises naturally in this context, and is discussed at the end of this chapter.