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Title: Asymptotic properties of the branching random walk
Author: Biggins, J. D.
ISNI:       0000 0001 3463 8764
Awarding Body: University of Oxford
Current Institution: University of Oxford
Date of Award: 1976
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The branching random walk is a Galton-Watson 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 Kesten-Stigum theorem for the Galton-Watson 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 age-dependent 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 RP 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)ΔC when the process survives where Δ is a suitable metric on the compact subsets of RP .(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:  DOI: Not available