Normed-convergence theory for supercritical branching processes

A proof is given of the basic normed-convergence theorem for the ordinary supercritical Bienaymé-Galton-Watson process with finite mean. Part of it is adapted to obtain an analogous result for inhomogeneous supercritical processes (i.e. branching processes in varying environment). This is used in part to give a detailed discussion on the normed- convergence behaviour of the ordinary process in the 'explosive' case (i.e with infinite mean); and rather pathological limit behaviour is found to obtain.