ВЛАСТИВОСТІ ВЕЛИКИХ ВІДХИЛЕНЬ ЕМПІРИЧНИХ ОЦІНОК У ЗАДАЧІ СТОХАСТИЧНОЇ ОПТИМІЗАЦІЇ ДЛЯ ОДНОРІДНОГО ВИПАДКОВОГО ПОЛЯ

• Datum: 2020
• Hauptverfasser: Knopov, P.S., Kasitskaya, E.J.
• Format: Artikel
• Sprache: English
• Veröffentlicht: V.M. Glushkov Institute of Cybernetics of NAS of Ukraine 2020
• Schlagworte: задача стохастичного програмування; однорідне у вузькому розумінні випадкове поле; умова сильного перемішування; великі відхилення
• Online Zugang: https://jais.net.ua/index.php/files/article/view/510
• Назва журналу: Problems of Control and Informatics

The paper is devoted to the consideration of a stochastic programming problem where the random factor is a homogeneous in a strict sense random field satisfying the strong mixing condition with the appropriate mixing coefficient. The first problem is approximated by the problem of minimization of the empirical function constructed on observations of the homogeneous random field. The conditions under which the empirical estimate is consistent are given, in particular, we have limits on the moments of the empirical function. When large deviations are investigated, some theorems from the functional analysis are used. In the theorems the estimate of large deviations of the empirical function from the former one implies the estimate of large deviations of the minimum point and the minimal value of the empirical function from analogous characteristics of the former problem. In particular, the notion of the conditioning function which describes the behavior of the minimized function in a neighborhood of a minimum point, is used. The fact that when the second argument is fixed, the function under the expectation sign can be considered as the element of the space of continuous functions, is used. For simplicity it assumes that the given function belongs to a convex compact subset of the appropriate functional space. The linear operators theory and the duality relation are used. In particular, the fact that a space of limited signed measures is dual to a space of continuous functions is used. The well-known results from large deviations theory, in particular, the theorem of a sufficient condition of the upper bound estimate for large deviations are applied. Notions of measurably separated random variables and the first hypermixing hypothesis for a homogeneous field are used. The auxiliary assertion on large deviations of empirical estimates in an abstract space is proved. In the proof the rectangle on the plane is divided on the subsets separated one from each other. Then the homogeneity of the field in a strict sense and the first hypermixing hypothesis imply the existence of the limit which is a sufficient condition for large deviations estimate.