# 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.

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

Other publications by J.D. Biggins