We investigate statistical properties of the optimal value of the Sample Average Approximation of stochastic programs, continuing the study (Krätschmer in Nonasymptotic upper estimates for errors of the sample average approximation method to solve risk averse stochastic programs, 2023. Forthcoming in SIAM J. Optim.). Central Limit Theorem type results are derived for the optimal value. As a crucial point the investigations are based on a new type of conditions from the theory of empirical processes which do not rely on pathwise analytical properties of the goal functions. In particular, continuity or convexity in the parameter is not imposed in advance as usual in the literature on the Sample Average Approximation method. It is also shown that the new condition is satisfied if the paths of the goal functions are Hölder continuous so that the main results carry over in this case. Moreover, the main results are applied to goal functions whose paths are piecewise Hölder continuous as e.g. in two stage mixed-integer programs. The main results are shown for classical risk neutral stochastic programs, but we also demonstrate how to apply them to the Sample Average Approximation of risk averse stochastic programs. In this respect we consider stochastic programs expressed in terms of absolute semideviations and divergence risk measures.

我们研究了随机程序的样本平均近似的最优值的统计特性，继续研究（Krätschmer，《解决风险厌恶随机程序的样本平均近似方法的误差的非渐近上限估计》，2023 年。即将发表于 SIAM J. Optim。 ）。中心极限定理类型的结果是针对最优值导出的。作为一个关键点，这些研究是基于经验过程理论中的一种新型条件，它不依赖于目标函数的路径分析特性。特别是，参数的连续性或凸性没有像样本平均近似方法的文献中通常那样预先强加。还表明，如果目标函数的路径是霍尔德连续的，则满足新条件，因此在这种情况下主要结果得以延续。此外，主要结果应用于路径是分段霍尔德连续的目标函数，例如在两阶段混合整数程序中。显示了经典风险中性随机程序的主要结果，但我们还演示了如何将它们应用于风险厌恶随机程序的样本平均近似。在这方面，我们考虑以绝对半偏差和分歧风险度量表示的随机程序。