Use this URL to cite or link to this record in EThOS:  http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.449795 
Title:  Asymptotic properties of the branching random walk  
Author:  Biggins, J. D.  
Awarding Body:  University of Oxford  
Current Institution:  University of Oxford  
Date of Award:  1976  
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Abstract:  
The branching random walk is a GaltonWatson process with the additional feature that people have positions. The initial ancestor is at the origin. Let {3^{(1)}_{r}} be the positions on the real line of his children. The people in the nth generation give birth independently of one another and of the preceeding generations to form the (n+1)th generation and the positions of the children of an nth generation person at x has the same distributions as {3^{(1)}_{r}+x} . Let {3^{(n)}_{r}} be positions of the nth generation people in this process. In the first chapter the convergence of certain martingales associated with this process is examined. A generalization of the KestenStigum theorem for the GaltonWatson process is obtained. The convergence of one of these martingales is shown to be closely related to some known results on the growth rate of agedependent branching process. If B^{(n)} is the position of the person on the extreme left of the nth generation then it is shown in the second chapter that B^{(n)}/n →γ for some constant γ when the process survives. Subsequent chapters are generalizations of this result. Thus the same result holds for a multitype process with a finite number of different types and a weaker result holds when there is a countable number of different types. The generalization to the branching random walk on R^{P} is also considered. Let C^{(n)} be the set of points {3^{(n)}_{r}/n:r}. It is shown that there is a compact convex set C such that C^{(n)}^{Δ}_{→}C when the process survives where Δ is a suitable metric on the compact subsets of R^{P} .(All of these results are proved under the 'natural' conditions).


Supervisor:  Not available  Sponsor:  Not available  
Qualification Name:  Thesis (Ph.D.)  Qualification Level:  Doctoral  
EThOS ID:  uk.bl.ethos.449795  DOI:  Not available  
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