Multitype branching processes based on exact progeny lengths of particles in a Galton-Watson branching process

In Gadag and Rajarshi, the authors studied a bivariate (multitype) branching process based on infinite and finite lines of descent, of particles of a supercritical one-dimensional (multitype) Galton-Watson branching process (GWBP). In this paper, they discuss a few more meaningful and interesting univariate and multitype branching processes, based on exact progeny lengths of particles in a GWBP. The present constructions relax the assumption of supercriticality made in Gadag and Rajarshi. The authors investigate some finite-time and asymptotic results of these processes in some details and relate them to the original process. These results are then used to propose new and better estimates of the offspring mean. An illustration based on the branching process of the white male population of the USA is also given. The authors believe that their work offers a rather finer understanding of the branching property.