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

by J.D.Biggins

*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 $\lambda$.

## Availability

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See also the closely related paper

Other publications by J.D. Biggins
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