Stochastic programming problems with probability and stochastic dominanceconstraints belong to "deterministic" problems depending on a probabilitymeasure. Complete knowledge of the probability measure is a necessary conditionto solve these problems. However since this assumption is fulfilled (in applications) very seldom, problems are mostly solved on a data base. Stochastic estimates of an optimal value and an optimal solution can be obtained then only. Properties of the estimates have been investigated many times, however mostly in the case of constraints sets not depending on the probability measure. Our results generalize them to two separate cases (mentioned already above) when the constraints sets depend on the probability measure. We focus on the case of heavy tailed distributions. First, we try to emphasize the results achieved for the above mentioned problems and the thin and heavy tailed distributions. However, the aim of the paper is mainly to analyze the approach to the problems of second order stochastic dominance constraints in the case of heavy tailed distributions. Namely, it seems that there can arise troubles that are not usual in the case of thin tails distributions. But the heavy tailed distributions (and specially stable distributions) correspond to many economic and financial applications, see e.g. [11], [12]. Theoretical analysis is completed by a simulation investigation.