# Convergence results in multitype, multivariate, branching random walk

by J.D.Biggins and A. Rahimzadeh Sani

2000 Mathematics Subject Classification: Primary 60J80 Secondary 15A48;60F10

## Abstract

A multi\-type branching random walk on $\dim$-dimensional
Euclidian space is considered. The uniform convergence, as $n$
goes to infinity, of a scaled version of the Laplace transform of
the point process given by the $n$th generation particles of each
type is obtained. Similar results in the one-type case, where the
transform gives a martingale, have been obtained in Biggins (1992)
and Barral (2001). This uniform convergence of transforms is then
used to obtain limit results for numbers in the underlying point
processes. Supporting results, which are of interest in their own
right, are obtained on (i) `Perron-Frobenius theory' for matrices
that are smooth functions of a variable $\lambda \in L$ and are
non-negative when
$\lambda \in L_{-}\subset L$ and (ii) saddlepoint approximations of multivariate
distributions. The saddlepoint approximations developed are strong
enough to give a refined large deviation theorem of Chaganty and
Sethuraman (1993) as a by-product.

## Availability:

Preprint 551/05,
Department of Probability and Statistics,
University of Sheffield. (January 2005). pdf file

minor revision (April 2005)

## To appear in: *Adv.
Appl. Probab.* (2005) **37**(3)

Other publications by J.D.
Biggins

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