Bøker av Dmitri (National Technical University of Ukraine (Igor Sikorsky Kyiv Polytechnic Institute)) Koroliouk

Within the field of modeling complex objects in natural sciences, which considers systems that consist of a large number of interacting parts, a good tool for analyzing and fitting models is the theory of random evolutionary systems, considering their asymptotic properties and large deviations. In Random Evolutionary Systems we consider these systems in terms of the operators that appear in the schemes of their diffusion and the Poisson approximation. Such an approach allows us to obtain a number of limit theorems and asymptotic expansions of processes that model complex stochastic systems, both those that are autonomous and those dependent on an external random environment. In this case, various possibilities of scaling processes and their time parameters are used to obtain different limit results.

This book is devoted to the system analysis of statistical experiments, determined by the averaged sums of sampling random variables. The dynamics of statistical experiments are given by difference stochastic equations with a speci?ed regression function of increments - linear or nonlinear. The statistical experiments are studied by the sample volume increasing (N ), as well as in discrete-continuous time by the number of stages increasing (k ) for different conditions imposed on the regression function of increments. The proofs of limit theorems employ modern methods for the operator and martingale characterization of Markov processes, including singular perturbation methods. Furthermore, they justify the representation of a stationary Gaussian statistical experiment with the Markov property, as a stochastic difference equation solution, applying the theorem of normal correlation. The statistical hypotheses verification problem is formulated in the classification of evolutionary processes, which determine the dynamics of the predictable component. The method of stochastic approximation is used for classifying statistical experiments.