Uniform convergence of martingales in the branching random walk
#
Uniform convergence of martingales in the branching random walk

by J.D.Biggins

*Ann. Probab*. (1992), **20**, 137-151.

1991 Mathematics Subject Classification: 60J80

## Abstract

In a discrete time supercritical branching random walk let \Zn be
the point process formed by the \nth\ generation. Let \ml be the
Laplace transform of the intensity measure of \Zo.
Then $\Wnl = \int e^{- \lambda x}
\Zn(dx)/\ml^{n}$,
which is the Laplace transform of \Zn normalized by its expected value,
forms a
martingale for any $\lambda$ with $|\ml|$ finite but non-zero.
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 \Wl is actually an
analytic function of \la.
The uniform convergence results are used
to obtain extensions of known results
on the growth of $\Zn(nc+D)$ with $n$, for bounded intervals $D$ and fixed $c$.
This forms the
second part of the paper, where
local large deviation results
for \Zn
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.

## Availability

postscript file
dvi file

See also the closely related paper

Other publications by J.D. Biggins
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