Random Geometry and Reinforced Jump Processes
(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 pseudotrajectories 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 pseudotrajectories 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 vertexreinforced jump processes (VRJP). Using the method of stochastic approximation, we show the connection between strongly VRJP and an asymptotic pseudotrajectory 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 historydependent random walk model in [0,1] based on Friedman’s urntype schemes and show that this random walk converges in distribution to the arcsine law (Less)
Please use this url to cite or link to this publication:
http://lup.lub.lu.se/record/1a0c3c3210a44207aba8ce3e164f1ce3
 author
 Nguyen, TuanMinh ^{LU}
 supervisor

 Stanislav Volkov ^{LU}
 Tatyana Turova ^{LU}
 opponent

 Professor Goldsheid, Ilya, School of Mathematical Sciences, Queen Mary University of London, United Kingdom
 organization
 publishing date
 201711
 type
 Thesis
 publication status
 published
 subject
 keywords
 random polygons, products of random matrices, vertexreinforced 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
 20171208 09:00
 ISBN
 9789177535065
 9789177535058
 language
 English
 LU publication?
 yes
 id
 1a0c3c3210a44207aba8ce3e164f1ce3
 date added to LUP
 20171109 10:53:00
 date last changed
 20171222 08:49:04
@phdthesis{1a0c3c3210a44207aba8ce3e164f1ce3, 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 pseudotrajectories 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 vertexreinforced jump processes (VRJP). Using the method of stochastic approximation, we show the connection between strongly VRJP and an asymptotic pseudotrajectory 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 historydependent random walk model in [0,1] based on Friedman’s urntype schemes and show that this random walk converges in distribution to the arcsine law}, author = {Nguyen, TuanMinh}, isbn = {9789177535065}, keyword = {random polygons,products of random matrices,vertexreinforced 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}, }