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Keywords:
stochastic programming problems; stability; Wasserstein metric; ${\cal L}_{1}$ norm; Lipschitz property; empirical estimates; convergence rate; exponential tails; heavy tails; Pareto distribution; risk functionals; empirical quantiles
stochastic programming problems; stability; Wasserstein metric; ${\cal L}_{1}$ norm; Lipschitz property; empirical estimates; convergence rate; exponential tails; heavy tails; Pareto distribution; risk functionals; empirical quantiles
Summary:
“Classical” optimization problems depending on a probability measure belong mostly to nonlinear deterministic optimization problems that are, from the numerical point of view, relatively complicated. On the other hand, these problems fulfil very often assumptions giving a possibility to replace the “underlying” probability measure by an empirical one to obtain “good” empirical estimates of the optimal value and the optimal solution. Convergence rate of these estimates have been studied mostly for “underlying” probability measures with suitable (thin) tails. However, it is known that probability distributions with heavy tails better correspond to many economic problems. The paper focuses on distributions with finite first moments and heavy tails. The introduced assertions are based on the stability results corresponding to the Wasserstein metric with an “underlying” $ {\cal L}_{1}$ norm and empirical quantiles convergence.
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