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January, 1992 Uniform Convergence of Martingales in the Branching Random Walk
J. D. Biggins
Ann. Probab. 20(1): 137-151 (January, 1992). DOI: 10.1214/aop/1176989921


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.


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J. D. Biggins. "Uniform Convergence of Martingales in the Branching Random Walk." Ann. Probab. 20 (1) 137 - 151, January, 1992.


Published: January, 1992
First available in Project Euclid: 19 April 2007

zbMATH: 0748.60080
MathSciNet: MR1143415
Digital Object Identifier: 10.1214/aop/1176989921

Primary: 60J80
Secondary: 60F10 , 60G42 , 60G44

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

Rights: Copyright © 1992 Institute of Mathematical Statistics

Vol.20 • No. 1 • January, 1992
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