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Random Geometry and Reinforced Jump Processes

Nguyen, Tuan-Minh LU (2017)
Abstract
This thesis comprises three papers studying several mathematical models related to geometric Markov processes and random processes with reinforcements. The main goal of these works is to investigate the dynamics as well as the limiting behaviour of the models as time goes to infinity, the existence of invariant measures and limiting distributions, the speed of convergence and other interesting relevant properties.
In the introduction, we firstly discuss the background: products of random matrices, asymptotic pseudo-trajectories and Markov chains in a general state space. We then outline motivation and overview of the main results in the papers included in this thesis.
In the first paper, we deal with a Markov chain model of convex... (More)
This thesis comprises three papers studying several mathematical models related to geometric Markov processes and random processes with reinforcements. The main goal of these works is to investigate the dynamics as well as the limiting behaviour of the models as time goes to infinity, the existence of invariant measures and limiting distributions, the speed of convergence and other interesting relevant properties.
In the introduction, we firstly discuss the background: products of random matrices, asymptotic pseudo-trajectories and Markov chains in a general state space. We then outline motivation and overview of the main results in the papers included in this thesis.
In the first paper, we deal with a Markov chain model of convex polygons, which are random consecutive subdivisions of an initial convex polygon. Applying the theory of products of random matrices, we prove the universal convergence of these random convex polygons to a “flat figure”. Beside this, we present a discussion about the speed of convergence and the computation of invariant measure in the case of random triangles.
In the second paper, we investigate a model of strongly vertex-reinforced jump processes (VRJP). Using the method of stochastic approximation, we show the connection between strongly VRJP and an asymptotic pseudo-trajectory of a vector field in order to study the dynamics of the model. In particular, we prove that strongly VRJP on a complete graph will almost surely have an infinite local time at one vertex, while the local times at all the remaining vertices remain bounded.
In the last paper, we consider a class of random walks taking values in simplexes and study the existence of limiting distributions. In some special cases of Markov chain models, we prove that the limiting distributions are Dirichlet. In addition, we introduce a related history-dependent random walk model in [0,1] based on Friedman’s urn-type schemes and show that this random walk converges in distribution to the arcsine law (Less)
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author
supervisor
opponent
  • Professor Goldsheid, Ilya, School of Mathematical Sciences, Queen Mary University of London, United Kingdom
organization
publishing date
type
Thesis
publication status
published
subject
keywords
random polygons, products of random matrices, vertex-reinforced jump processes, pseudotrajectories, random walks in simplexes, Markov chains in a general state space
pages
148 pages
publisher
Lund University, Faculty of Science, Centre for Mathematical Sciences, Mathematical Statistics
defense location
Lecture hall MH:R, Matematikcentrum, Sölvegatan 18A, Lund
defense date
2017-12-08 09:00
ISBN
978-91-7753-505-8
978-91-7753-506-5
language
English
LU publication?
yes
id
1a0c3c32-10a4-4207-aba8-ce3e164f1ce3
date added to LUP
2017-11-09 10:53:00
date last changed
2017-12-22 08:49:04
@phdthesis{1a0c3c32-10a4-4207-aba8-ce3e164f1ce3,
  abstract     = {This thesis comprises three papers studying several mathematical models related to geometric Markov processes and random processes with reinforcements. The main goal of these works is to investigate the dynamics as well as the limiting behaviour of the models as time goes to infinity, the existence of invariant measures and limiting distributions, the speed of convergence and other interesting relevant properties.<br/>In the introduction, we firstly discuss the background: products of random matrices, asymptotic pseudo-trajectories and Markov chains in a general state space. We then outline motivation and overview of the main results in the papers included in this thesis.<br/>In the first paper, we deal with a Markov chain model of convex polygons, which are random consecutive subdivisions of an initial convex polygon. Applying the theory of products of random matrices, we prove the universal convergence of these random convex polygons to a “flat figure”. Beside this, we present a discussion about the speed of convergence and the computation of invariant measure in the case of random triangles.<br/>In the second paper, we investigate a model of strongly vertex-reinforced jump processes (VRJP). Using the method of stochastic approximation, we show the connection between strongly VRJP and an asymptotic pseudo-trajectory of a vector field in order to study the dynamics of the model. In particular, we prove that strongly VRJP on a complete graph will almost surely have an infinite local time at one vertex, while the local times at all the remaining vertices remain bounded.<br/>In the last paper, we consider a class of random walks taking values in simplexes and study the existence of limiting distributions. In some special cases of Markov chain models, we prove that the limiting distributions are Dirichlet. In addition, we introduce a related history-dependent random walk model in [0,1] based on Friedman’s urn-type schemes and show that this random walk converges in distribution to the arcsine law},
  author       = {Nguyen, Tuan-Minh},
  isbn         = {978-91-7753-505-8},
  keyword      = {random polygons,products of random matrices,vertex-reinforced jump processes,pseudotrajectories, random walks in simplexes,Markov chains in a general state space},
  language     = {eng},
  pages        = {148},
  publisher    = {Lund University, Faculty of Science, Centre for Mathematical Sciences, Mathematical Statistics},
  school       = {Lund University},
  title        = {Random Geometry and Reinforced Jump Processes},
  year         = {2017},
}