The Annals of Probability

Uniform Convergence of Martingales in the Branching Random Walk

J. D. Biggins

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Abstract

In a discrete-time supercritical branching random walk, let $Z^{(n)}$ be the point process formed by the $n$th generation. Let $m(\lambda)$ be the Laplace transform of the intensity measure of $Z^{(1)}$. Then $W^{(n)}(\lambda) = \int e^{-\lambda x}Z^{(n)}(dx)/m(\lambda)^n$, which is the Laplace transform of $Z^{(n)}$ normalized by its expected value, forms a martingale for any $\lambda$ with $|m(\lambda)|$ finite but nonzero. The convergence of these martingales uniformly in $\lambda$, for $\lambda$ lying in a suitable set, is the first main result of this paper. This will imply that, on that set, the martingale limit $W(\lambda)$ is actually an analytic function of $\lambda$. The uniform convergence results are used to obtain extensions of known results on the growth of $Z^{(n)}(nc + D)$ with $n$, for bounded intervals $D$ and fixed $c$. This forms the second part of the paper, where local large deviation results for $Z^{(n)}$ which are uniform in $c$ are considered. Finally, similar results, both on martingale convergence and uniform local large deviations, are also obtained for continuous-time models including branching Brownian motion.

Article information

Source
Ann. Probab., Volume 20, Number 1 (1992), 137-151.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176989921

Digital Object Identifier
doi:10.1214/aop/1176989921

Mathematical Reviews number (MathSciNet)
MR1143415

Zentralblatt MATH identifier
0748.60080

JSTOR
links.jstor.org

Subjects
Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60F10: Large deviations 60G42: Martingales with discrete parameter 60G44: Martingales with continuous parameter

Keywords
Spatial growth in branching processes uniform local large deviations Banach space valued martingales

Citation

Biggins, J. D. Uniform Convergence of Martingales in the Branching Random Walk. Ann. Probab. 20 (1992), no. 1, 137--151. doi:10.1214/aop/1176989921. https://projecteuclid.org/euclid.aop/1176989921


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