Asymptotic properties of the branching random walk

Thesis

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 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.)

Authors: Biggins, J

• DOI: 10.5287/ora-bdzjpeqvq
• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: University of Oxford