ИСТИНА

Random fields are often used as building blocks of various stochastic models. We consider the problems related to the analysis of random fields trajectories and the asymptotic problems concerning the excursion sets when the windows of observations are growing and the level (or levels) determining these sets is fixed or tends to infinity. Such problems arise, e.g., in material sciences when one tries to investigate the local structure of matter. Here we prove and employ the central limit theorem for specified functionals in random fields. The dependence structure of random fields under consideration plays the key role. The approach developed in by A.V. Bulinski and A.P. Shashkin in 2007 permits to study not only the Gaussian random fields. The approximations of initial random fields by auxiliary ones are also widely applied. After that we will concentrate on the study of models having applications in genetics. In this research domain we are interested in spatial models describing the relationship between the genotype and phenotype (the environmental explanatory variables are considered as well). Such problems are important for identification of the genetic factors which could increase the risk of complex disease. In this way we prove the results providing the base for dimensionality reduction of observations and discuss the problems of the model selection, optimal in a sense. Several approaches involving cross-validation, random graphs, trees and forests, logic regression and its modifications are in the scope of the talk. Simulation techniques is also tackled. The final part of the talk is devoted to stochastic models using the spatial point processes. Here we treat the statistical problems related to the population dynamics. Beyond the survey of the state of art and the new established results some open problems are discussed as well.