Uniform convergence of martingales in the one-dimensional branching random walk.

# Uniform convergence of martingales in the one-dimensional branching random walk.

IMS lectures notes -- Monograph Series. Selected proceedings of the Sheffield Symposium on Applied Probability, 1989, Eds. I.V. Basawa and R.L. Taylor. (1991), 18, 159-173.

1991 Mathematics Subject Classification: 60J80

## Abstract

In the supercritical branching random walk an initial person has children whose positions are given by a point process \Zo. Each of these then has children in the same way, with the positions of children in each family, relative to their parent's, being given by independent copies of \Zo, and so on. For any value of its argument, $\lambda$, the Laplace transform of the point process of \nth\ generation people, normalized by its expected value, is a martingale, the usual branching process martingale being a special case. Here it is shown that under certain conditions these martingales converge uniformly in $\lambda$, almost surely and in mean. A consequence of this result is that the limit is, in an appropriate region, analytic in $\lambda$.

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