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Some aspects of tree-indexed processes
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Sheehan, Marcus (2006) Some aspects of tree-indexed processes. PhD thesis, University of Warwick.
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Official URL: http://webcat.warwick.ac.uk/record=b3759433
Abstract
Chapters 2 and 3 of this thesis are based on a paper in preparation, “Partial observations of a tree-indexed process”. We begin, in Chapter 1, with an introduction to the theory of branching random walks. We then consider stochastic process indexed by Galton-Watson trees and highlight the connection between “speeds” for the corresponding branching random walks and large deviation theory. We relate all of this to a class of martingales obtained via a change of measure corresponding to changing the distribution of the tree- indexed random variables along a randomly chosen “line of descent” through the tree. We show that the issue of uniform integrability for these martingales - a much studied subject in its own right - boils down essentially to large deviation calculations.
In Chapter 2 we introduce “recovery problems”. We observe some information about a tree-indexed collection of random variables and ask if it is possible to “recover” the original random variables in some suitable sense. We motivate this with a simple example on the integers and then describe the analogous problem on the binary tree, using the theory developed in Chapter 1 to derive conditions on the underlying probability parameters under which recovery is possible.
In Chapter 3 we turn our attention to “Recursive Distributional Equations” (or RDEs). Motivated by the work of Chapter 2 we investigate a recursion for random variables on the binary tree and then study the corresponding RDE in its own right by introducing the idea of tree-indexed solutions. We are able to give a fairly complete analysis of the RDE in the case where recovery of the tree-indexed random variables from Chapter 2 is possible but find that the non-recovery case is less tractable. Here we provide partial results and make conjectures based on what we know to be true.
In Chapter 4 we further develop the theory of RDEs by introducing the notion of “en- dogeny”. This relates to whether (tree-indexed) solutions to RDEs can be written as functions of the original data alone or whether there is some additional randomness coming from the system. We conclude the chapter with a particular example, the so-called “noisy veto voter model”, and obtain conditions for endogeny in this setting by extending some recent work in this area.
Chapter 5 is based on a paper in preparation, “A recursive distributional equation on [0,1]”. The RDE in question is obtained from the noisy veto voter RDE of Chapter 4 via various transformations and conditioning. We make a thorough study of this new RDE by identifying all invariant distributions, the corresponding “basins of attraction” and addressing the issue of endogeny for the associated tree-indexed problem.
At the end of each chapter we discuss briefly possible extensions to the work and make clear any unresolved issues or simplifying assumptions that might be relaxed in future research.
Item Type: | Thesis (PhD) | ||||
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Subjects: | Q Science > QA Mathematics | ||||
Library of Congress Subject Headings (LCSH): | Branching processes, Stochastic processes, Random walks (Mathematics), Martingales (Mathematics), Recursive functions | ||||
Official Date: | May 2006 | ||||
Dates: |
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Institution: | University of Warwick | ||||
Theses Department: | Department of Statistics | ||||
Thesis Type: | PhD | ||||
Publication Status: | Unpublished | ||||
Supervisor(s)/Advisor: | Warren, Jon (Reader in statistics) ; Jacka, Saul D. | ||||
Sponsors: | Engineering and Physical Sciences Research Council | ||||
Format of File: | |||||
Extent: | 163 leaves | ||||
Language: | eng |
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