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Dissertations / Theses on the topic 'Random processes'

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Published: 4 June 2021

Last updated: 9 June 2025

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1

Makai, Tamas. "Random graph processes." Thesis, Royal Holloway, University of London, 2012. http://repository.royalholloway.ac.uk/items/b24b89af-3fc1-4d2f-a673-64483a3bc2f2/8/.

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This thesis deals with random graph processes. More precisely it deals with two random graph processes which create H -free graphs. The first of these processes is the random H-elimination process which starts from the complete graph and in every step removes an edge uniformly at random from the set of edges which are found in a copy of H. The second is the H-free random graph process which starts from the empty graph and in every step an edge chosen uniformly at random from the set of edges which when added to the graph would not create a copy of H is inserted. We consider these graph processes for several classes of graphs H, for example strictly two balanced graphs. The class of strictly two balanced graphs includes among others cycles and complete graphs. We analysed the H-elimination process, when H is strictly 2-balanced. For this class we show the typical number of edges found at the end of the process. We also consider the sub graphs created by the process and its independence number. We also managed to show the expected number of edges in the H -elimination pro- cess when H = Ki, the graph created from the complete graph on 4 vertices by removing an edge and when H = K34 where K34 is created from the complete bi- partite graph with 3 vertices in one partition and'4 vertices in the second partition, by removing an edge. In case of the H -free process we considered the case when H is the triangle and showed that the triangle-free random graph process only creates sparse subgraphs. Finally we have improved the lower bound on the length of the K34-free random graph process. '

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2

Kang, Mihyun. "Random planar structures and random graph processes." Doctoral thesis, [S.l.] : [s.n.], 2007. http://deposit.ddb.de/cgi-bin/dokserv?idn=985516585.

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3

Zielen, Frank H. "Asymmetric random average processes." [S.l.] : [s.n.], 2002. http://deposit.ddb.de/cgi-bin/dokserv?idn=965270475.

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4

Timar, Adam. "Group-invariant random processes." [Bloomington, Ind.] : Indiana University, 2006. http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&res_dat=xri:pqdiss&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&rft_dat=xri:pqdiss:3204537.

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Thesis (Ph.D.)--Indiana University, Dept. of Mathematics, 2006.<br>Source: Dissertation Abstracts International, Volume: 67-01, Section: B, page: 0308. Adviser: Russell Lyons. "Title from dissertation home page (viewed Feb. 9, 2007)."

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Ortgiese, Marcel. "Stochastic processes in random environment." Thesis, University of Bath, 2009. https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.507234.

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We are interested in two probabilistic models of a process interacting with a random environment. Firstly, we consider the model of directed polymers in random environment. In this case, a polymer, represented as the path of a simple random walk on a lattice, interacts with an environment given by a collection of time-dependent random variables associated to the vertices. Under certain conditions, the system undergoes a phase transition from an entropy-dominated regime at high temperatures, to a localised regime at low temperatures. Our main result shows that at high temperatures, even though a central limit theorem holds, we can identify a set of paths constituting a vanishing fraction of all paths that supports the free energy. We compare the situation to a mean-field model defined on a regular tree, where we can also describe the situation at the critical temperature. Secondly, we consider the parabolic Anderson model, which is the Cauchy problem for the heat equation with a random potential. Our setting is continuous in time and discrete in space, and we focus on time-constant, independent and identically distributed potentials with polynomial tails at infinity. We are concerned with the long-term temporal dynamics of this system. Our main result is that the periods, in which the profile of the solutions remains nearly constant, are increasing linearly over time, a phenomenon known as ageing. We describe this phenomenon in the weak sense, by looking at the asymptotic probability of a change in a given time window, and in the strong sense, by identifying the almost sure upper envelope for the process of the time remaining until the next change of profile. We also prove functional scaling limit theorems for profile and growth rate of the solution of the parabolic Anderson model.

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6

Warnke, Lutz. "Random graph processes with dependencies." Thesis, University of Oxford, 2012. http://ora.ox.ac.uk/objects/uuid:71b48e5f-a192-4684-a864-ea9059a25d74.

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Random graph processes are basic mathematical models for large-scale networks evolving over time. Their systematic study was pioneered by Erdös and Rényi around 1960, and one key feature of many 'classical' models is that the edges appear independently. While this makes them amenable to a rigorous analysis, it is desirable, both mathematically and in terms of applications, to understand more complicated situations. In this thesis the main goal is to improve our rigorous understanding of evolving random graphs with significant dependencies. The first model we consider is known as an Achlioptas process: in each step two random edges are chosen, and using a given rule only one of them is selected and added to the evolving graph. Since 2000 a large class of 'complex' rules has eluded a rigorous analysis, and it was widely believed that these could give rise to a striking and unusual phenomenon. Making this explicit, Achlioptas, D'Souza and Spencer conjectured in Science that one such rule yields a very abrupt (discontinuous) percolation phase transition. We disprove this, showing that the transition is in fact continuous for all Achlioptas process. In addition, we give the first rigorous analysis of the more 'complex' rules, proving that certain key statistics are tightly concentrated (i) in the subcritical evolution, and (ii) also later on if an associated system of differential equations has a unique solution. The second model we study is the H-free process, where random edges are added subject to the constraint that they do not complete a copy of some fixed graph H. The most important open question for such 'constrained' processes is due to Erdös, Suen and Winkler: in 1995 they asked what the typical final number of edges is. While Osthus and Taraz answered this in 2000 up to logarithmic factors for a large class of graphs H, more precise bounds are only known for a few special graphs. We close this gap for the cases where a cycle of fixed length is forbidden, determining the final number of edges up to constants. Our result not only establishes several conjectures, it is also the first which answers the more than 15-year old question of Erdös et. al. for a class of forbidden graphs H.

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7

Ding, Xinhong Carleton University Dissertation Mathematics. "Diffusion processes with random interactions." Ottawa, 1992.

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8

Seierstad, Taral Guldahl. "The phase transition in random graphs and random graph processes." Doctoral thesis, [S.l.] : [s.n.], 2007. http://deposit.ddb.de/cgi-bin/dokserv?idn=985760044.

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9

Hu, Yilei. "Essays on random processes with reinforcement." Thesis, University of Oxford, 2010. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.669991.

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10

Figwer, Jarosław. "Synthesis and simulation of random processes." Praca habilitacyjna, Wydawnictwo Politechniki Śląskiej, 1999. https://delibra.bg.polsl.pl/dlibra/docmetadata?showContent=true&id=8070.

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11

Hannigan, Patrick. "Random polynomials." Thesis, University of Ulster, 1998. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.263248.

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12

Blanc, Guillaume. "Poisson processes of roads and random geometry." Electronic Thesis or Diss., université Paris-Saclay, 2024. http://www.theses.fr/2024UPASM024.

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Le thème général de cette thèse est la géométrie aléatoire.Nous commençons par étudier la métrique aléatoire construite par Kendall à partir d'un processus de Poisson de routes auto-similaire dans l'espace euclidien (par routes, nous entendons droites avec des limitations de vitesse).Dans un premier temps, nous nous intéressons aux propriétés fractales de cet espace métrique aléatoire, qui est presque sûrement homéomorphe à l'espace euclidien, et dont nous calculons la dimension de Hausdorff comme une constante strictement plus grande que la dimension ambiante.Dans un autre travail, nous considérons les propriétés des géodésiques pour cette métrique aléatoire, en montrant en particulier que dans le cas planaire, celles-ci ne font pas de pause en route (c'est-à-dire, n'utilisent pas de routes de vitesse arbitrairement petite en dehors de leurs extrémités), conformément à une conjecture de Kendall. En étudiant de manière analogue une métrique aléatoire construite à partir d'un processus de Poisson de routes dans un arbre régulier, nous considérons le phénomène d'explosion et montrons que celui-ci présente une transition de phase : en fonction du paramètre qui régit les limitations de vitesse des routes, il est possible ou non de rouler vers l'infini en temps fini.Indépendamment, nous étudions le modèle de coloriage poissonnien dans l'espace euclidien, et nous montrons que la frontière dans ce modèle est fractale, au sens où sa dimension de Hausdorff est comprise strictement entre la dimension ambiante moins un et la dimension ambiante.Cela confirme une conjecture d'Aldous. Enfin, nous nous intéressons aux distances entre points typiques dans les espaces métriques mesurés, en se demandant quelles lois sont la fonction à deux points d'un espace métrique mesuré<br>The general subject of this thesis is random geometry.We begin by studying the random metric constructed by Kendall from a self-similar Poisson process of roads (i.e, lines with speed limits) in Euclidean space.First, we consider the fractal properties of this random metric space: almost surely, it is homeomorphic to the ambient Euclidean space, and we calculate its Hausdorff dimension to be a constant strictly greater than the ambient dimension.Furthermore, we study the geodesics for this random metric, and prove in particular that in the planar case, they do not pause en route (i.e, use roads with arbitrarily low speed away from their endpoints), thus confirming a conjecture of Kendall.Analogously, we study a random metric constructed from a Poisson process of roads in a regular tree, and consider the explosion phenomenon, which we show undergoes a phase transition: depending on the parameter which governs the speed limits of the roads, it is possible or not to drive to infinity in finite time.Independently, we consider the Poissonian colouring model in Euclidean space, and prove that the frontier in this model is fractal, in the sense that its Hausdorff dimension lies strictly between the ambient dimension minus one and the ambient dimension.This confirms a conjecture of Aldous.Finally, we study distances between typical points in measured metric spaces, by investigating which distributions arise as the two-point function of a measured metric space

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13

Yeo, Dominic. "Self-organised criticality in random graph processes." Thesis, University of Oxford, 2016. https://ora.ox.ac.uk/objects/uuid:23af1abc-2128-4315-9b25-55ed8f290875.

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In the first half of this thesis, we study the random forest obtained by conditioning the Erdös-Rényi random graph G(N,p) to include no cycles. We focus on the critical window, in which <style> td.upper_line { border-top:solid 1px black; } table.fraction { text-align: center; vertical-align: middle; margin-top:0.5em; margin-bottom:0.5em; line-height: 2em; } </style> <table class="fraction" align="center" cellpadding="0" cellspacing="0"> <tr> <td rowspan="2" nowrap="nowrap"> <i>p(N)</i> = </td><td nowrap="nowrap"> 1+λN<sup>-1/3</sup> </td> </tr><tr> <td class="upper_line"> N </td> </tr> </table> , as studied by Aldous for G(N,p). We describe a scaling limit for the sizes of the largest trees in this critical random forest, in terms of the excursions above zero of a particular reflected diffusion. We proceed by showing convergence of the reflected exploration process associated to the critical random forests, using careful enumeration of classes of forests, and the asymptotic properties of uniform trees. In the second half of this thesis, we study a random graph process where vertices have one of k types. An inhomogeneous random graph represents the initial connections between vertices, and over time new edges are added homogeneously, as in the classical random graph process. Each vertex is frozen at some rate, resulting in the removal of its entire component. This is a version of the frozen percolation model introduced by R\'ath, which (under mild conditions) exhibits self-organised criticality: the dynamics first drive the system to a critical state, and from then on maintain it in criticality. We prove a convergence result for the proportion of vertices of each type which survive until time t, and describe the local limit in terms of a multitype branching process whose parameters are critical and given by the solution to an unusual differential equation driven by Perron--Frobenius eigenvectors. The argument relies on a novel multitype exploration process, leading to a concentration result for the proportion of types in all large components of a near-critical inhomogeneous random graph; and on a stronger convergence result for mean-field frozen percolation, when the initial graphs may be random.

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14

Jones, Elinor Mair. "Large deviations of random walks and levy processes." Thesis, University of Manchester, 2008. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.491853.

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15

Kandler, Anne, Matthias Richter, Scheidt Jürgen vom, Hans-Jörg Starkloff, and Ralf Wunderlich. "Moving-Average approximations of random epsilon-correlated processes." Universitätsbibliothek Chemnitz, 2004. http://nbn-resolving.de/urn:nbn:de:swb:ch1-200401266.

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The paper considers approximations of time-continuous epsilon-correlated random processes by interpolation of time-discrete Moving-Average processes. These approximations are helpful for Monte-Carlo simulations of the response of systems containing random parameters described by epsilon-correlated processes. The paper focuses on the approximation of stationary epsilon-correlated processes with a prescribed correlation function. Numerical results are presented.

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16

Schmid, Patrick. "Random processes in truncated and ordinary Weyl chambers." Doctoral thesis, Universitätsbibliothek Leipzig, 2011. http://nbn-resolving.de/urn:nbn:de:bsz:15-qucosa-66394.

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The work consists of two parts. In the first part which is concerned with random walks, we construct the conditional versions of a multidimensional random walk given that it does not leave the Weyl chambers of type C and of type D, respectively, in terms of a Doob h-transform. Furthermore, we prove functional limit theorems for the rescaled random walks. This is an extension of recent work by Eichelsbacher and Koenig who studied the analogous conditioning for the Weyl chamber of type A. Our proof follows recent work by Denisov and Wachtel who used martingale properties and a strong approximation of random walks by Brownian motion. Therefore, we are able to keep minimal moment assumptions. Finally, we present an alternate function that is amenable to an h-transform in the Weyl chamber of type C. In the second part which is concerned with Brownian motion, we examine the non-exit probability of a multidimensional Brownian motion from a growing truncated Weyl chamber. Different regimes are identified according to the growth speed, ranging from polynomial decay over stretched-exponential to exponential decay. Furthermore we derive associated large deviation principles for the empirical measure of the properly rescaled and transformed Brownian motion as the dimension grows to infinity. Our main tool is an explicit eigenvalue expansion for the transition probabilities before exiting the truncated Weyl chamber.

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17

Shykula, Mykola. "Quantization of Random Processes and Related Statistical Problems." Doctoral thesis, Umeå : Department of Mathematics and Mathematical Statistics, Umeå University, 2006. http://urn.kb.se/resolve?urn=urn:nbn:se:umu:diva-883.

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18

Padgett, Wayne Thomas. "Detection of low order nonstationary gaussian random processes." Diss., Georgia Institute of Technology, 1994. http://hdl.handle.net/1853/13523.

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19

Jones, Owen Dafydd. "Random walks on pre-fractals and branching processes." Thesis, University of Cambridge, 1995. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.388440.

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20

Smith, Jason Marko. "Discrete properties of continuous, non-Gaussian random processes." Thesis, University of Nottingham, 2007. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.438330.

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21

El-Feghi, Farag Abdulrazzak. "Miscible flooding in correlated random fields." Thesis, Heriot-Watt University, 1992. http://hdl.handle.net/10399/1506.

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22

Buckley, Stephen Philip. "Problems in random walks in random environments." Thesis, University of Oxford, 2011. http://ora.ox.ac.uk/objects/uuid:06a12be2-b831-4c2a-87b1-f0abccfb9b8b.

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Recent years have seen progress in the analysis of the heat kernel for certain reversible random walks in random environments. In particular the work of Barlow(2004) showed that the heat kernel for the random walk on the infinite component of supercritical bond percolation behaves in a Gaussian fashion. This heat kernel control was then used to prove a quenched functional central limit theorem. Following this work several examples have been analysed with anomalous heat kernel behaviour and, in some cases, anomalous scaling limits. We begin by generalizing the first result - looking for sufficient conditions on the geometry of the environment that ensure standard heat kernel upper bounds hold. We prove that these conditions are satisfied with probability one in the case of the random walk on continuum percolation and use the heat kernel bounds to prove an invariance principle. The random walk on dynamic environment is then considered. It is proven that if the environment evolves ergodically and is, in a certain sense, geometrically d-dimensional then standard on diagonal heat kernel bounds hold. Anomalous lower bounds on the heat kernel are also proven - in particular the random conductance model is shown to be "more anomalous" in the dynamic case than the static. Finally, the reflected random walk amongst random conductances is considered. It is shown in one dimension that under the usual scaling, this walk converges to reflected Brownian motion.

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23

Naumov, Alexey [Verfasser]. "Universality of some models of random matrices and random processes / Alexey Naumov. Fakultät für Mathematik." Bielefeld : Universitätsbibliothek Bielefeld, Hochschulschriften, 2013. http://d-nb.info/1032454121/34.

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24

Ling, Hong. "Implementation of Stochastic Neural Networks for Approximating Random Processes." Master's thesis, Lincoln University. Environment, Society and Design Division, 2007. http://theses.lincoln.ac.nz/public/adt-NZLIU20080108.124352/.

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Artificial Neural Networks (ANNs) can be viewed as a mathematical model to simulate natural and biological systems on the basis of mimicking the information processing methods in the human brain. The capability of current ANNs only focuses on approximating arbitrary deterministic input-output mappings. However, these ANNs do not adequately represent the variability which is observed in the systems natural settings as well as capture the complexity of the whole system behaviour. This thesis addresses the development of a new class of neural networks called Stochastic Neural Networks (SNNs) in order to simulate internal stochastic properties of systems. Developing a suitable mathematical model for SNNs is based on canonical representation of stochastic processes or systems by means of Karhunen-Loève Theorem. Some successful real examples, such as analysis of full displacement field of wood in compression, confirm the validity of the proposed neural networks. Furthermore, analysis of internal workings of SNNs provides an in-depth view on the operation of SNNs that help to gain a better understanding of the simulation of stochastic processes by SNNs.

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25

Grobys, Sebastian. "Random generation of Bayesian Networks and Markov Decision Processes." Thesis, McGill University, 2004. http://digitool.Library.McGill.CA:80/R/?func=dbin-jump-full&object_id=81336.

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Markov Decision Processes (MDPs) and Bayesian Networks (BNs) are two very different but equally prominent frameworks for modeling stochastic environments. As empirical testing and the use of randomized search algorithms gain increasing importance in machine learning and artificial intelligence, there exists a growing need for random models; in particular MDPs and BNs. This thesis aims at providing a focused review of the field of random model generation. We begin by motivating in detail the need for rigorous random MDP and BN generation and survey relevant past research efforts. We outline the different problems involved in generating the models and then present our main contribution, a random graph algorithm specifically designed for the uniform generation of graph structures underlying random MDPs and BNs.

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26

Tokushige, Yuki. "Random Walks on random trees and hyperbolic groups: trace processes on boundaries at infinity and the speed of biased random walks." Kyoto University, 2019. http://hdl.handle.net/2433/242580.

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27

Hägg, Jonas. "Gaussian fluctuations in some determinantal processes." Doctoral thesis, KTH, Matematik (Inst.), 2007. http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-4343.

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This thesis consists of two parts, Papers A and B, in which some stochastic processes, originating from random matrix theory (RMT), are studied. In the first paper we study the fluctuations of the kth largest eigenvalue, xk, of the Gaussian unitary ensemble (GUE). That is, let N be the dimension of the matrix and k depend on N in such a way that k and N-k both tend to infinity as N - ∞. The main result is that xk, when appropriately rescaled, converges in distribution to a Gaussian random variable as N → ∞. Furthermore, if k1 < ...< km are such that k1, ki+1 - ki and N - km, i =1, ... ,m - 1, tend to infinity as N → ∞ it is shown that (xk1 , ... , xkm) is multivariate Gaussian in the rescaled N → ∞ limit. In the second paper we study the Airy process, A(t), and prove that it fluctuates like a Brownian motion on a local scale. We also prove that the Discrete polynuclear growth process (PNG) fluctuates like a Brownian motion in a scaling limit smaller than the one where one gets the Airy process.<br>QC 20100716

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28

Kang, Sungyeol. "Extreme values of random times in stochastic networks." Diss., Georgia Institute of Technology, 1992. http://hdl.handle.net/1853/24305.

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29

Hermann, Felix [Verfasser], and Peter [Akademischer Betreuer] Pfaffelhuber. "On dualities of random graphs and branching processes with disasters to Piecewise deterministic Markov processes." Freiburg : Universität, 2019. http://d-nb.info/1182225985/34.

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30

Yu, Jihnhee. "Approaches to the multivariate random variables associated with stochastic processes." Texas A&M University, 2003. http://hdl.handle.net/1969.1/1209.

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Stochastic compartment models are widely used in modeling processes for biological populations. The residence time has been especially useful in describing the system dynamics in the models. The direct calculation of the distribution for the residence time of stochastic multi-compartment models is very complicated even with a relatively simple model and often impossible to calculate directly. This dissertation presents an analytical method to obtain the moment generating function for stochastic multi-compartment models and describe the distribution of the residence times, especially systems with nonexponential lifetime distributions. A common method for obtaining moments of the residence time is using the coefficient matrix, however it has a limitation in obtaining high order moments and moments for combined compartments in a system. In this dissertation, we first derive the bivariate moment generating function of the residence time distribution for stochastic two-compartment models with general lifetimes. It provides any order of moments and also enables us to approximate the density of the residence time using the saddlepoint approximation. The approximation method is applied to various situations including the approximation of the bivariate distribution of residence times in two-compartment models or approximations based on the truncated moment generating function. Special attention is given to the distribution of the residence time for multi-compartment semi-Markov models. The cofactor rule and the analytic approach to the two-compartment model facilitate the derivation of the moment generating function. The properties from the embedded Markov chain are also used to extend the application of the approach. This approach provides a complete specification of the residence time distribution based on the moment generating function and thus provides an easier calculation of high-order moments than the approach using the coefficient matrix. Applications to drug kinetics demonstrate the simplicity and usefulness of this approach.

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Frings, René [Verfasser]. "Interlacing Patterns in Exclusion Processes and Random Matrices / René Frings." Bonn : Universitäts- und Landesbibliothek Bonn, 2014. http://d-nb.info/1047622793/34.

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32

Gabrysch, Katja. "On Directed Random Graphs and Greedy Walks on Point Processes." Doctoral thesis, Uppsala universitet, Analys och sannolikhetsteori, 2016. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-305859.

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This thesis consists of an introduction and five papers, of which two contribute to the theory of directed random graphs and three to the theory of greedy walks on point processes. We consider a directed random graph on a partially ordered vertex set, with an edge between any two comparable vertices present with probability p, independently of all other edges, and each edge is directed from the vertex with smaller label to the vertex with larger label. In Paper I we consider a directed random graph on ℤ2 with the vertices ordered according to the product order and we show that the limiting distribution of the centered and rescaled length of the longest path from (0,0) to (n, [na] ), a<3/14, is the Tracy-Widom distribution. In Paper II we show that, under a suitable rescaling, the closure of vertex 0 of a directed random graph on ℤ with edge probability n−1 converges in distribution to the Poisson-weighted infinite tree. Moreover, we derive limit theorems for the length of the longest path of the Poisson-weighted infinite tree. The greedy walk is a deterministic walk on a point process that always moves from its current position to the nearest not yet visited point. Since the greedy walk on a homogeneous Poisson process on the real line, starting from 0, almost surely does not visit all points, in Paper III we find the distribution of the number of visited points on the negative half-line and the distribution of the index at which the walk achieves its minimum. In Paper IV we place homogeneous Poisson processes first on two intersecting lines and then on two parallel lines and we study whether the greedy walk visits all points of the processes. In Paper V we consider the greedy walk on an inhomogeneous Poisson process on the real line and we determine sufficient and necessary conditions on the mean measure of the process for the walk to visit all points.

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33

Belu, Alexandru C. "Multivariate Measures of Dependence for Random Variables and Levy Processes." Case Western Reserve University School of Graduate Studies / OhioLINK, 2012. http://rave.ohiolink.edu/etdc/view?acc_num=case1333396376.

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34

Mogharehabed, Saeideh. "A Suboptimal Multi-Clustering Algorithm for Random Labeled Point Processes." The Ohio State University, 2017. http://rave.ohiolink.edu/etdc/view?acc_num=osu1483010792818099.

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Fu, Zuopeng. "Karlin Random Fields: Limit Theorems, Representations and Simulations." University of Cincinnati / OhioLINK, 2020. http://rave.ohiolink.edu/etdc/view?acc_num=ucin1613754836854037.

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36

Fedorov, Valery V., and Werner Müller. "Optimum design for correlated processes via eigenfunction expansions." Institut für Statistik und Mathematik, WU Vienna University of Economics and Business, 2004. http://epub.wu.ac.at/622/1/document.pdf.

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In this paper we consider optimum design of experiments for correlated observations. We approximate the error component of the process by an eigenvector expansion of the corresponding covariance function. Furthermore we study the limit behavior of an additional white noise as a regularization tool. The approach is illustrated by some typical examples. (authors' abstract)<br>Series: Research Report Series / Department of Statistics and Mathematics

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Abramowicz, Konrad. "Numerical analysis for random processes and fields and related design problems." Doctoral thesis, Umeå universitet, Institutionen för matematik och matematisk statistik, 2011. http://urn.kb.se/resolve?urn=urn:nbn:se:umu:diva-46156.

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In this thesis, we study numerical analysis for random processes and fields. We investigate the behavior of the approximation accuracy for specific linear methods based on a finite number of observations. Furthermore, we propose techniques for optimizing performance of the methods for particular classes of random functions. The thesis consists of an introductory survey of the subject and related theory and four papers (A-D). In paper A, we study a Hermite spline approximation of quadratic mean continuous and differentiable random processes with an isolated point singularity. We consider a piecewise polynomial approximation combining two different Hermite interpolation splines for the interval adjacent to the singularity point and for the remaining part. For locally stationary random processes, sequences of sampling designs eliminating asymptotically the effect of the singularity are constructed. In Paper B, we focus on approximation of quadratic mean continuous real-valued random fields by a multivariate piecewise linear interpolator based on a finite number of observations placed on a hyperrectangular grid. We extend the concept of local stationarity to random fields and for the fields from this class, we provide an exact asymptotics for the approximation accuracy. Some asymptotic optimization results are also provided. In Paper C, we investigate numerical approximation of integrals (quadrature) of random functions over the unit hypercube. We study the asymptotics of a stratified Monte Carlo quadrature based on a finite number of randomly chosen observations in strata generated by a hyperrectangular grid. For the locally stationary random fields (introduced in Paper B), we derive exact asymptotic results together with some optimization methods. Moreover, for a certain class of random functions with an isolated singularity, we construct a sequence of designs eliminating the effect of the singularity. In Paper D, we consider a Monte Carlo pricing method for arithmetic Asian options. An estimator is constructed using a piecewise constant approximation of an underlying asset price process. For a wide class of Lévy market models, we provide upper bounds for the discretization error and the variance of the estimator. We construct an algorithm for accurate simulations with controlled discretization and Monte Carlo errors, andobtain the estimates of the option price with a predetermined accuracy at a given confidence level. Additionally, for the Black-Scholes model, we optimize the performance of the estimator by using a suitable variance reduction technique.

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38

Jones, Brian Douglas. "Tree components in random graph processes with non-uniform edge probabilities /." The Ohio State University, 1995. http://rave.ohiolink.edu/etdc/view?acc_num=osu1487867541733461.

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39

Strang, Alexander. "Applications of the Helmholtz-Hodge Decomposition to Networks and Random Processes." Case Western Reserve University School of Graduate Studies / OhioLINK, 2020. http://rave.ohiolink.edu/etdc/view?acc_num=case1595596768356487.

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40

Oosthuizen, Joubert. "Random walks on graphs." Thesis, Stellenbosch : Stellenbosch University, 2014. http://hdl.handle.net/10019.1/86244.

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Thesis (MSc)--Stellenbosch University, 2014.<br>ENGLISH ABSTRACT: We study random walks on nite graphs. The reader is introduced to general Markov chains before we move on more specifically to random walks on graphs. A random walk on a graph is just a Markov chain that is time-reversible. The main parameters we study are the hitting time, commute time and cover time. We nd novel formulas for the cover time of the subdivided star graph and broom graph before looking at the trees with extremal cover times. Lastly we look at a connection between random walks on graphs and electrical networks, where the hitting time between two vertices of a graph is expressed in terms of a weighted sum of e ective resistances. This expression in turn proves useful when we study the cover cost, a parameter related to the cover time.<br>AFRIKAANSE OPSOMMING: Ons bestudeer toevallige wandelings op eindige gra eke in hierdie tesis. Eers word algemene Markov kettings beskou voordat ons meer spesi ek aanbeweeg na toevallige wandelings op gra eke. 'n Toevallige wandeling is net 'n Markov ketting wat tyd herleibaar is. Die hoof paramaters wat ons bestudeer is die treftyd, pendeltyd en dektyd. Ons vind oorspronklike formules vir die dektyd van die verdeelde stergra ek sowel as die besemgra ek en kyk daarna na die twee bome met uiterste dektye. Laastens kyk ons na 'n verband tussen toevallige wandelings op gra eke en elektriese netwerke, waar die treftyd tussen twee punte op 'n gra ek uitgedruk word in terme van 'n geweegde som van e ektiewe weerstande. Hierdie uitdrukking is op sy beurt weer nuttig wanneer ons die dekkoste bestudeer, waar die dekkoste 'n paramater is wat verwant is aan die dektyd.

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Peng, Man Kallenberg Olav. "Palm measure invariance and exchangeability for marked point processes." Auburn, Ala, 2008. http://repo.lib.auburn.edu/EtdRoot/2008/FALL/Mathematics_and_Statistics/Dissertation/Peng_Man_3.pdf.

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42

Breen, Barbara J. "Computational nonlinear dynamics monostable stochastic resonance and a bursting neuron model /." Diss., Available online, Georgia Institute of Technology, 2004:, 2003. http://etd.gatech.edu/theses/available/etd-04082004-180036/unrestricted/breen%5Fbarbara%5Fj%5F200312%5Fphd.pdf.

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43

Van, Zyl Alexis J. "Basic concepts of random matrix theory." Thesis, Stellenbosch : University of Stellenbosch, 2005. http://hdl.handle.net/10019.1/1624.

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Thesis (MSc (Physics))--University of Stellenbosch, 2005.<br>It was Wigner that in the 1950’s first introduced the idea of modelling physical reality with an ensemble of random matrices while studying the energy levels of heavy atomic nuclei. Since then, the field of Random Matrix Theory has grown tremendously, with applications ranging from fluctuations on the economic markets to M-theory. It is the purpose of this thesis to discuss the basic concepts of Random Matrix Theory, using the ensembles of random matrices originally introduced by Wigner, the Gaussian ensembles, as a starting point. As Random Matrix Theory is classically concerned with the statistical properties of levels sequences, we start with a brief introduction to the statistical analysis of a level sequence before getting to the introduction of the Gaussian ensembles. With the ensembles defined, we move on to the statistical properties that they predict. In the light of these predictions, a few of the classical applications of Random Matrix Theory are discussed, and as an example of some of the important concepts, the Anderson model of localization is investigated in some detail.

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44

Wong, Ka Yiu. "Model-free tests for isotropy, equal distribution and random superposition in spatial point pattern analysis." HKBU Institutional Repository, 2015. https://repository.hkbu.edu.hk/etd_oa/202.

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This thesis introduces three new model-free tests for isotropy, equal distribution and random superposition in non-rectangular windows respectively. For isotropy, a bootstrap-type test is proposed. The corresponding test statistic assesses the discrepancy between the uniform distribution and the empirical normalised reduced second-order moment measure of a sector of fixed radius with increasing central angle. The null distribution of the discrepancy is then estimated by stochastic reconstruction, which generates bootstrap-type samples of point patterns that resemble the spatial structure of the given pattern. The new test is applicable for small sample sizes and is shown to have more robust powers to different choices of user-chosen parameter when compared with the asymptotic chi-squared test by Guan et al. (2006) in our simulation. For equal distribution, a model-free asymptotic test is introduced. The proposed test statistic compares the discrepancy between the empirical second-order product densities of the observed point patterns at some pre-chosen lag vectors. Under certain mild moment conditions and a weak dependence assumption, the limiting null distribution of the test statistic is the chi-squared distribution. Simulation results show that the new test is more powerful than the permutation test by Hahn (2012) for comparing point patterns with similar structures but different distributions. The new test for random superposition is a modification of the toroidal shift test by Lotwick and Silverman (1982). The idea is to extrapolate the pattern observed in a non-rectangular window to a larger rectangular region by the stochastic reconstruction so that the toroidal shift test can be applied. Simulation results show that the powers of the test applied to patterns with extrapolated points are remarkably higher than those of the test applied to the largest inscribed rectangular windows, with only slightly increased type I error rates. Real data sets are used to illustrate the advantages of the tests developed in this thesis over the existing tests in the literature.

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45

Greenberg, Sam. "Random sampling of lattice configurations using local Markov chains." Diss., Atlanta, Ga. : Georgia Institute of Technology, 2008. http://hdl.handle.net/1853/28090.

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Thesis (M. S.)--Mathematics, Georgia Institute of Technology, 2009.<br>Committee Chair: Randall, Dana; Committee Member: Heitsch, Christine; Committee Member: Mihail, Milena; Committee Member: Trotter, Tom; Committee Member: Vigoda, Eric.

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46

Vo, Ba Tuong. "Random finite sets in Multi-object filtering." University of Western Australia. School of Electrical, Electronic and Computer Engineering, 2008. http://theses.library.uwa.edu.au/adt-WU2009.0045.

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[Truncated abstract] The multi-object filtering problem is a logical and fundamental generalization of the ubiquitous single-object vector filtering problem. Multi-object filtering essentially concerns the joint detection and estimation of the unknown and time-varying number of objects present, and the dynamic state of each of these objects, given a sequence of observation sets. This problem is intrinsically challenging because, given an observation set, there is no knowledge of which object generated which measurement, if any, and the detected measurements are indistinguishable from false alarms. Multi-object filtering poses significant technical challenges, and is indeed an established area of research, with many applications in both military and commercial realms. The new and emerging approach to multi-object filtering is based on the formal theory of random finite sets, and is a natural, elegant and rigorous framework for the theory of multiobject filtering, originally proposed by Mahler. In contrast to traditional approaches, the random finite set framework is completely free of explicit data associations. The random finite set framework is adopted in this dissertation as the basis for a principled and comprehensive study of multi-object filtering. The premise of this framework is that the collection of object states and measurements at any time are treated namely as random finite sets. A random finite set is simply a finite-set-valued random variable, i.e. a random variable which is random in both the number of elements and the values of the elements themselves. Consequently, formulating the multiobject filtering problem using random finite set models precisely encapsulates the essence of the multi-object filtering problem, and enables the development of principled solutions therein. '...' The performance of the proposed algorithm is demonstrated in simulated scenarios, and shown at least in simulation to dramatically outperform traditional single-object filtering in clutter approaches. The second key contribution is a mathematically principled derivation and practical implementation of a novel algorithm for multi-object Bayesian filtering, based on moment approximations to the posterior density of the random finite set state. The performance of the proposed algorithm is also demonstrated in practical scenarios, and shown to considerably outperform traditional multi-object filtering approaches. The third key contribution is a mathematically principled derivation and practical implementation of a novel algorithm for multi-object Bayesian filtering, based on functional approximations to the posterior density of the random finite set state. The performance of the proposed algorithm is compared with the previous, and shown to appreciably outperform the previous in certain classes of situations. The final key contribution is the definition of a consistent and efficiently computable metric for multi-object performance evaluation. It is shown that the finite set theoretic state space formulation permits a mathematically rigorous and physically intuitive construct for measuring the estimation error of a multi-object filter, in the form of a metric. This metric is used to evaluate and compare the multi-object filtering algorithms developed in this dissertation.

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47

Altay, Suhan. "On Forward Interest Rate Models: Via Random Fields And Markov Jump Processes." Master's thesis, METU, 2007. http://etd.lib.metu.edu.tr/upload/12608342/index.pdf.

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The essence of the interest rate modeling by using Heath-Jarrow-Morton framework is to find the drift condition of the instantaneous forward rate dynamics so that the entire term structure is arbitrage free. In this study, instantaneous forward interest rates are modeled using random fields and Markov Jump processes and the drift conditions of the forward rate dynamics are given. Moreover, the methodology presented in this study is extended to certain financial settings and instruments such as multi-country interest rate models, term structure of defaultable bond prices and forward measures. Also a general framework for bond prices via nuclear space valued semi-martingales is introduced.

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48

Puplinskaitė, Donata. "Aggregation of autoregressive processes and random fields with finite or infinite variance." Doctoral thesis, Lithuanian Academic Libraries Network (LABT), 2013. http://vddb.library.lt/obj/LT-eLABa-0001:E.02~2013~D_20131029_102339-22917.

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Aggregated data appears in many areas such as econimics, sociology, geography, etc. This motivates an importance of studying the (dis)aggregation problem. One of the most important reasons why the contemporaneous aggregation become an object of research is the possibility of obtaining the long memory phenomena in processes. The aggregation provides an explanation of the long-memory effect in time series and a simulation method of such series as well. Accumulation of short-memory non-ergodic random processes can lead to the long memory ergodic process, that can be used for the forecasts of the macro and micro variables. We explore the aggregation scheme of AR(1) processes and nearest-neighbour random fields with infinite variance. We provide results on the existence of limit aggregated processes, and find conditions under which it has long memory properties in certain sense. For the random fields on Z^2, we introduce the notion of (an)isotropic long memory based on the behavior of partial sums. In L_2 case, the known aggregation of independent AR(1) processes leads to the Gaussian limit. While we describe a new model of aggregation based on independent triangular arrays. This scheme gives the limit aggregated process with finite variance which is not necessary Gaussian. We study a discrete time risk insurance model with stationary claims, modeled by the aggregated heavy-tailed process. We establish the asymptotic properties of the ruin probability and the dependence structure... [to full text]<br>Agreguoti duomenys naudojami daugelyje mokslo sričių tokių kaip ekonomika, sociologija, geografija ir kt. Tai motyvuoja tirti (de)agregavimo uždavinį. Viena iš pagrindinių priežasčių kodėl vienalaikis agregavimas tapo tyrimų objektu yra galimybė gauti ilgos atminties procesus. Agregavimas paaiškina ilgos atminties atsiradima procesuose ir yra vienas iš būdų tokius procesus generuoti. Agreguodami trumpos atminties neergodiškus atsitiktinius procesus, galime gauti ilgos atminties ergodišką procesą, kuris gali būti naudojamas mikro ir makro kintamųjų prognozavimui. Disertacijoje nagrinėjama AR(1) procesų bei artimiausio kaimyno atsitiktinių laukų, turinčių begalinę dispersiją, agregavimo schema, randamos sąlygos, kurioms esant ribinis agreguotas procesas egzistuoja, ir turi ilgąją atmintį tam tikra prasme. Atsitiktinių laukų atveju, įvedamas anizotropinės/izotropinės ilgos atminties apibrėžimas, kuris yra paremtas dalinių sumų elgesiu. Baigtinės dispersijos atveju yra gerai žinoma nepriklausomų AR(1) procesų schema, kuri rezultate duoda Gauso ribinį agreguotą procesą. Disertacijoje aprašoma trikampio masyvo agregavimo modelis, kuris baigtinės dispersijos atveju duoda nebūtinai Gauso ribinį agreguotą procesą. Taip pat disertacijoje nagrinėjama bankroto tikimybės asimptotika, kai žalos yra aprašomos sunkiauodegiu agreguotu procesu, nusakoma priklausomybė tarp žalų, apibūdinama žalų ilga atmintis.

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49

Bernergård, Zandra. "Connection between discrete time random walks and stochastic processes by Donsker's Theorem." Thesis, Mälardalens högskola, Akademin för utbildning, kultur och kommunikation, 2020. http://urn.kb.se/resolve?urn=urn:nbn:se:mdh:diva-48719.

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In this paper we will investigate the connection between a random walk and a continuous time stochastic process. Donsker's Theorem states that a random walk under certain conditions will converge to a Wiener process. We will provide a detailed proof of this theorem which will be used to prove that a geometric random walk converges to a geometric Brownian motion.

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50

Fellenberg, Benno, Jürgen vom Scheidt, and Matthias Richter. "Simulation of Weakly Correlated Functions and its Application to Random Surfaces and Random Polynomials." Universitätsbibliothek Chemnitz, 1998. http://nbn-resolving.de/urn:nbn:de:bsz:ch1-199801258.

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The paper is dedicated to the modeling and the simulation of random processes and fields. Using the concept and the theory of weakly correlated functions a consistent representation of sufficiently smooth random processes will be derived. Special applications will be given with respect to the simulation of road surfaces in vehicle dynamics and to the confirmation of theoretical results with respect to the zeros of random polynomials.

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