Journal Article

**serial: **Acta Mathematica Universitas Comenianae vol.84, 2 (2015), p. 267-281

**project(s): **GA13-14445S, GA ČR

**keywords: **Stochastic programming problems,
empirical estimates,
light and heavy tailed distributions,
quantiles

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**abstract (eng): **

Stochastic optimization problems with an operator of the mathematical expectation in the objective function, probability and stochastic dominance constraints belong to “deterministic” problems depending on a probability measure. Complete knowledge of the probability measure is a necessary condition for solving these problems. However, since this assumption is very rarely fulfilled (in applications), problems are mostly solved on the basis of data. Mathematically it means that the “underlying” probability measure is replaced by an empirical one (determined by the corresponding data). Stochastic estimates of an optimal value and an optimal solution can only then be obtained. Properties of these estimates have been investigated many times, mostly in the case of constraint sets not depending on the probability measure. Our results generalize such estimates to two separate cases (already mentioned above) when the constraint sets do depend on the probability measure.

**RIV: **BB