Conceptual proofs of \(L\log L\) criteria for mean behavior of branching processes. (English) Zbl 0840.60077

Lyons, Russell; Pemantle, Robin; Peres, Yuval

Ann. Probab. 23, No. 3, 1125-1138 (1995).

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 [H. Kesten and 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 [C. R. Heathcote, E. Seneta and 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 [H. Kesten, P. Ney and F. Spitzer, ibid. 11, 513-540 (1966); resp. ibid. 11, 579-611 (1966; Zbl 0158.35202)].

Reviewer: D.R.Grey (Sheffield)

MSC:

60J80

Branching processes (Galton-Watson, birth-and-death, etc.)

Keywords:

size biased distributions; Galton-Watson process; Kesten-Stigum theorem; exponential limit law

Citations:

Zbl 0185.44902; Zbl 0203.17401; Zbl 0166.14202; Zbl 0158.35202