## The Annals of Probability

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

### Uniform Convergence of Martingales in the Branching Random Walk

#### 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