Heavy tails, extremes and clusters of linear stochastic recursive sequences

In the dissertation we study the linear stochastic recursive sequences of the first order. We introduce a new class of distributions, called Brownian mixtures and study their properties. We propose a new approach to linear stochastic recurrent sequences associated with the studied process that satisfies the stochastic differential equation, observed at random intervals. We find the different boundaries of extreme index for the case of Brownian mixtures. We prove the theorem of the continuous dependence of the tail index, extreme index and size distributions of clusters on distribution of coefficients. We obtain exact results in the case of ternary and generalized Laplace distributions. We consider a generalization of the tail and extreme indices to the multidimensional case with applications to vector-matrix stochastic difference equations.