首页|Properties of Large Deviations of Empirical Estimates in a Stochastic Optimization Problem for a Homogeneous Random Field
Properties of Large Deviations of Empirical Estimates in a Stochastic Optimization Problem for a Homogeneous Random Field
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The paper is devoted to the consideration of a stochastic programming problem where the random factor is homogeneous in a strict sense random field satisfying the strong mixing condition with the appropriate 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, it is imposed restrictions on the moments of the minimized 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 concept of 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 operator's 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 the 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 into the subsets separated 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.
stochastic programming problema homogeneous in a strict sense random fieldstrong mixing conditionlarge deviations
P.S. Knopov、E.J. Kasitskaya
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National Academy of Sciences of Ukraine, V. M. Giushkov Institute of Cybernetics of National Academy of Sciences of Ukraine, Kiev
V.M. Giushkov Institute of Cybernetics of National Academy of Sciences of Ukraine, Kiev
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