Article,
Growth rates in the branching random walk
Z. Wahrsch. Verw. Gebiete, 48 (1): 17--34 (1979)
Abstract
Consider a Galton-Watson process with offspring mean $m>1$, where the individuals have positions on the line. The position of the children, relative to the position of their mother, forms a general point process $\xi$ on the line. If $F(t)=E\xi(-ınfty,t$, then $F^n^\ast$ describes the intensity measure of the $n$th generation, and the behaviour of $F^n^\ast(nt)$ has earlier been described by the author J. Appl. Prob. 14 (1977), no. 1, 25--37; MR0433619 (55 #6592), thereby generalizing classical results of Chernoff, and Bahadur and Ranga Rao on sums of independent random variables. The present paper deals with stochastic analogues, viz., the proof of existence of nondegenerate limits of $Z_n(nb)/F^n^\ast(nb)$, where $Z_n(nb)$ is the number of particles at time $n$ to the left of $nb$. The proofs are probabilistic and based on martingales and the additivity property of branching processes.
