ON THE APPLICATIONS OF REGULARLY VARYING FUNCTIONS IN THE THEORY OF BRANCHING PROCESSES

We consider a discrete-time Galton-Vatson branching process. Our main analytical tool is the slow variation (or more general, a regular variation)conception in the sense of Karamata. The slow variation property arises in many issues, but it usually remains rather hidden. Application of Karamata functions in the branching processes theory allows one to bypass severe constraints concerning existence of the higher-order moments of the infinitesimal characteristics of the process under study. In this work,delving deeply in the nature of the Karamata functions, we study more subtle properties of branching processes.