Limit theorems for two classes of Markov processes

Abstract

The thesis includes two Parts that analyse the asymptotic behaviour of two multicomponent stochastic processes. In both cases, the components of the processes are highly dependent, however dynamics of two processes are signi cantly di erent. Part I is devoted to the study of hierarchical models with local dependence: behaviour of the (i + 1)'st component is in uenced by the i'th component only. These processes are either null-recurrent or transient and, therefore, do not possess limiting distributions. We analyse the structure of these processes and obtain limit theorems under normalisation. Part II deals with another type of models that arise in the neural systems. We consider symmetric models: any permutation of the coordinates has the same type of dynamics. We analyse the structure of these processes and conditions for positive recurrence.