#
Seneta-Heyde norming in the branching random walk

by J.D.Biggins and
A.E.Kyprianou ).

*Ann. Probab*. (1997) **25** 337-360.

1991 Mathematics Subject Classification: 60J80

## Abstract

In the discrete-time
supercritical branching random walk there is a Kesten-Stigum
type result for the martingales formed by the Laplace
transform of the $n$th generation positions. Roughly, this
says that for suitable values of the argument of the Laplace transform
the martingales converge in mean provided an $X \log X$ condition
holds. Here it is established that when this moment condition
fails, so that the martingale converges to zero,
it is possible to find a
(Seneta-Heyde) renormalization of the martingale
that converges
(in probability) to a finite non-zero limit when the process survives.
As part of the proof a Seneta-Heyde renormalization of
the general (C-M-J) branching process is obtained; in this case
the convergence holds almost surely. The results rely
heavily on a detailed study of the
functional equation that the Laplace transform of
the limit must satisfy.

## Availability

postscript file
dvi file

See also the companion paper

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