Empirical Estimates in Optimization Problems: Survey with Special Regard to Heavy Tails and Dependent Sample

<p><p><span style="font-size: 11.000000pt; font-family: 'CMR10';">Economic processes are usually influenced simultaneously by a random factor and a decision parameter. Since the decision parameter has to be mostly determined before realization of the random element, deterministic optimization problems which depend on a probability measure often correspond to the above mentioned situations. A complete knowledge of the “underlying” measure would be a necessary assumption to determine both an exact optimal solution and an exact optimal value. Since this condition is not usually fulfilled, the solution is often determined on an empirical data base. Corresponding estimates can only be obtained using this approach. </span> <span style="font-size: 11.000000pt; font-family: 'CMR10';">Many efforts have been made to investigate the above mentioned estimates. The consis- tency, convergence rate and an asymptotic distribution have been examined. This was mostly done under assumptions of linear dependence on the probability measure, distri- butions with “thin” tails and an assumption of independent data. The aim of this paper is to consider the cases in which these assumptions are rather relaxed. To this end we employ stability results based on the Wasserstein metric corresponding to </span><span style="font-size: 11.000000pt; font-family: 'CMSY10';">L</span><span style="font-size: 8.000000pt; font-family: 'CMR8'; vertical-align: -2.000000pt;">1 </span><span style="font-size: 11.000000pt; font-family: 'CMR10';">norm and some results on mixing sequences. </span>