Conceptual proofs of \(L\log L\) criteria for mean behavior of branching processes (Q1902944)

A Galton-Watson process is constructed on the space of family trees, and then a related process is constructed on the same space by giving one individual in each generation a family size whose distribution is the size-biased version of the original family size distribution. Using a constant interplay between the two probability measures representing these two processes, and a number of ingenious technical devices, new and insightful proofs are obtained of three classical theorems in the theory of the Galton-Watson process: the Kesten-Stigum theorem on normed growth in the supercritical case [\textit{H. Kesten} and \textit{B. O. Stigum}, Ann. Math. Stat. 37, 1211-1223 (1966; Zbl 0203.17401)]; the rate of decay of probability of non-extinction in the subcritical case [\textit{C. R. Heathcote}, \textit{E. Seneta} and \textit{D. Vere-Jones}, Theory Probab. Appl. 12(1967), 297-301 (1968) and Teor. Veroyatn. Primen. 12, 341-346 (1967; Zbl 0166.14202)]; and the rate of decay of probability of non-extinction and normed exponential limit law conditional on non-extinction in the critical case [\textit{H. Kesten}, \textit{P. Ney} and \textit{F. Spitzer}, ibid. 11, 513-540 (1966); resp. ibid. 11, 579-611 (1966; Zbl 0158.35202)].