Article,
Chernoff's theorem in the branching random walk
J. Appl. Probability, 14 (3): 630--636 (1977)
Abstract
Consider a supercritical Galton-Watson process with the additional structure of the individuals having positions on the line. The number and positions of the children of an individual at $x$ are described by a point process centered around $x$, and the point processes corresponding to different individuals are i.i.d. The author obtains geometric estimates of the rate of growth of the number $Z^(n)(na)$ of individuals in $(-ınfty,na)$ at time $n$. The results contain as a corollary results on the first birth problem for the Bellman-Harris process given by J. F. C. Kingman Ann. Probability 3 (1975), no. 5, 790--801; MR0400438 (53 #4271) and generalized to the branching random walk by the author Advances in Appl. Probability 8 (1976), no. 3, 446--459; MR0420890 (54 #8901). As usual in branching processes, the problem is closely related to the study of the mean $EZ^(n)(na)$, which in the present setting is along the lines of the work of H. Chernoff Ann. Math. Statist. 23 (1952), 493--507; MR0057518 (15,241c) and R. R. Bahadur and R. Ranga Rao ibid. 31 (1960), 1015--1027; MR0117775 (22 #8549) dealing with geometric estimates for the probabilities in the weak law.
