From Asymptotic Normality to Heavy-Tailedness via Limit Theorems for Random Sums and Statistics with Random Sample Sizes

This chapter contains a possible explanation of the emergence of heavy-tailed distributions observed in practice instead of the expected normal laws. The bases for this explanation are limit theorems for random sums and statistics constructed from samples with random sizes. As examples of the application of general theorems, conditions are presented for the convergence of the distributions of random sums of independent random vectors with finite covariance matrices to multivariate elliptically contoured stable and Linnik distributions. Also, conditions are presented for the convergence of the distributions of asymptotically normal (in the traditional sense) statistics to multivariate Student distributions. The joint asymptotic behavior of sample quantiles is also considered

Probabilistic Methods for Cognitive Solving of Some Problems in Artificial Intelligence Systems

As a result of the analysis of dispatcher intelligence centers and aerial, land, underground, underwater, universal, and functionally focused artificial intelligence robotics systems, the problems of rational control, due to be performed under specific conditions of uncertainties, are chosen for probabilistic study. The choice covers the problems of planning the possibilities of functions performance on the base of monitored information about events and conditions and the problem of robot route optimization under limitations on risk of “failure” in conditions of uncertainties. These problems are resolved with a use of the proposed probabilistic approach. The proposed methods are based on selected probabilistic models (for “black box” and complex systems), which are implemented effectively in wide application areas. The cognitive solving of problems consists in improvements, accumulation, analysis, and use of appearing knowledge. The described analytical solutions are demonstrated by practical examples

Two approaches to the construction of perturbation bounds for continuous-time Markov chains

The paper is largely of a review nature. It considers two main methods used to study stability and obtain appropriate quantitative estimates of perturbations of (inhomogeneous) Markov chains with continuous time and a finite or countable state space. An approach is described to the construction of perturbation estimates for the main five classes of such chains associated with queuing models. Several specific models are considered for which the limit characteristics and perturbation bounds for admissible "perturbed" processes are calculated