Orientador: Serguei Popov / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matemática Estatística e Computação Científica / Made available in DSpace on 2018-08-25T09:42:00Z (GMT). No. of bitstreams: 1 Costa_MarceloRocha_M.pdf: 576974 bytes, checksum: 3b01a9f15e6e0f9fdd98631dc69cd202 (MD5) Previous issue date: 2014 / Resumo: O problema considerado no presente trabalho cumpre o papel de reforçar a eficácia dos métodos apresentados nos capítulos introdutórios, bem como investiga a resposta de um problema até então não publicado na literatura especializada. Introduzimos uma partícula realizando um passeio aleatório simples no espaço, ou seja, uma partícula que a cada passo escolhe uniformemente um de seus vizinhos para onde irá saltar. Fixando sua posição inicial ao longo da fronteira do cubo, pergunta-se: qual é a probabilidade de que a trajetória de tal partícula nunca mais retorne ao cubo? Em outras palavras, se T é o tempo de primeiro retorno ao cubo, estamos interessados em descrever o comportamento assintótico da probabilidade de que T seja infinito / Abstract: It has been considered in this work a problem which play a role of showing the effectiveness of the content covered in the introductory chapters, as well as it is a unsolved problem across the specialized literature. We introduce a particle performing a simple random walk in space, i.e., a particle which at each step choose uniformly one of its neighbourhood sites to which it then jumps into. Fixed its initial position along the boundary of a cube, we are interested in answering the following question: what is the probability that such particle's trajectory will never reach the cube again. In other words, if T is the first return time to the cube, we aim to analyse the asymptotic behaviour of the probability that T is infinite / Mestrado / Estatistica / Mestre em Estatística

Orientador: Serguei Popov / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática Estatística e Computação Científica / Made available in DSpace on 2018-08-25T10:22:43Z (GMT). No. of bitstreams: 1 Bernardini_DiegoFernandode_D.pdf: 1410086 bytes, checksum: b77a17aefd06d547f1c5db3c5cc1a8f7 (MD5) Previous issue date: 2014 / Resumo: O modelo de entrelaçamentos aleatórios foi introduzido no sentido de se investigar originalmente o traço deixado por passeios aleatórios em grandes grafos e, basicamente, tal processo é descrito por um processo pontual de Poisson em um espaço de trajetórias duplamente infinitas de passeios aleatórios simples no reticulado d-dimensional, com dimensão d pelo menos igual a três. Neste sentido, o processo é caracterizado por um emaranhado aleatório de trajetórias deste tipo. Tal modelo possui ainda um parâmetro de intensidade, que controla, de certa forma, a quantidade de trajetórias que constituem o processo. Um problema relevante no contexto deste processo, e que tem sido amplamente estudado na literatura, diz respeito à caracterização da relação de dependência (através da covariância) entre os eventos denominados como crescentes neste modelo e suportados em subconjuntos disjuntos do reticulado, e é justamente este o problema no qual nos concentramos. Em uma primeira etapa neste trabalho, determinamos expressões explícitas para a derivada, com respeito ao parâmetro de intensidade, da probabilidade de um evento crescente e suportado em um subconjunto finito do reticulado, estabelecendo assim aquilo que denominamos como a fórmula de Russo para os entrelaçamentos aleatórios. A utilização desta denominação é justificada e motivada pelo amplamente conhecido termo original, que no contexto do modelo usual de percolação estabelece uma expressão para a derivada da probabilidade dos eventos definidos como crescentes naquele modelo. Em seguida, tentamos utilizar este resultado no sentido de estabelecer uma primeira abordagem para o problema da covariância entre os eventos crescentes, e esta investigação é baseada essencialmente em uma observação sobre o número esperado das trajetórias então denominadas como pivotais positivas para o evento de interesse. Por fim, estabelecemos uma nova abordagem para o mesmo problema, utilizando uma construção alternativa do processo de entrelaçamentos baseada na técnica dos soft local times, e investigando uma espécie de pivotalidade conjunta de coleções de excursões das trajetórias dos passeios aleatórios pelos conjuntos nos quais estão suportados os eventos de interesse. Justamente a partir desta abordagem obtemos nosso último resultado sobre a covariância. De forma geral, acreditamos que a investigação e a tentativa de obter uma caracterização cada vez mais precisa para a relação de dependência que mencionamos deve ajudar a entender o processo de entrelaçamentos e suas propriedades de forma cada vez mais clara / Abstract: The random interlacements model was originally introduced in order to investigate the trace left by random walks in large graphs and, basically, such process is described by a Poisson point process in a space of doubly infinite simple random walk trajectories in the d-dimensional lattice, with dimension d at least equal to three. In this sense, the process is characterized by a random tangle of trajectories of this kind. Such model also has an intensity parameter, which controls, in a certain sense, the quantity of trajectories that constitutes the process. A relevant issue in the context of this process, which has been largely studied in the literature, concerns the characterization of the dependence relation (through the covariance) between the so-called increasing events in this model, which are supported on disjoint subsets of the lattice, and this is precisely the issue on which we focus. In a first step in this work, we determine explicit expressions for the derivative, with respect to the intensity parameter, of the probability of an increasing event which is supported in a finite subset of the lattice, thus establishing what we call as Russo¿s formula for random interlacements. The use of this term is justified and motivated by the widely known original term, which, in the context of the usual percolation model, provides an expression for the derivative of the probability of events defined as increasing in that model. Then, we try to use this result to establish a first approach to the problem of the covariance between increasing events, and such investigation is essentially based in a fact about the expected number of the so-called positive pivotal (or plus pivotal) trajectories for the event of interest. Finally, we establish a new approach to the same problem by using an alternative construction of the interlacements process based on the technique of soft local times, and investigating a kind of joint "pivotality" of collections of excursions of the random walk trajectories, through the sets on which the events of interest are supported. From this approach we obtain our last result on the covariance. Overall, we believe that the investigation and the attempt to get an increasingly accurate characterization of the above mentioned dependence relation should help to understand the interlacements process and its properties in an increasingly clear way / Doutorado / Estatistica / Doutor em Estatística

This paper deals with the joint and conditional distributions concerning the maximum of random walk paths and the number of times this maximum is achieved. This joint distribution was studied first by Dwass [1967]. Based on his result, the correlation and some conditional moments are derived. The main contributions are however asymptotic expansions concerning the conditional distribution and conditional moments. (author's abstract) / Series: Forschungsberichte / Institut für Statistik

The ratchet effect can be exploited in many types of research, yet few researchers pay attention to it. In this thesis, I investigate two examples of such effects in microfluidic devices, under the guidance of computational simulations. The first chapter provides a brief introduction to ratchet effects, electrophoresis, and swimming cells, topics directly related to the following chapters. The second chapter of this thesis studies the separation of charged spherical particles in various microfluidic devices. My work shows how to manipulate those particles with modified temporal asymmetric electric potentials. The rectification of randomly swimming bacteria in microfluidic devices has been extensively studied. However, there have been few attempts to optimize such rectification devices. Mapping such motion onto a lattice Monte Carlo model may suggest some new mathematical methods, which might be useful for optimizing the similar systems. Such a mapping process is introduced in chapter four.

Les dispersions colloïdales ont beaucoup d’applications technologiques importantes. A cause du mouvement brownien, les particules ont des collisions fréquentes entre elles. Les forces d’attraction de van der Waals,dérivant de potentiels à longue portés, conduisent à l’agrégation et à la précipitation des particules. Plusieurs méthodes ont été proposées pour diminuer ou contrebalancer l’effet d’attraction de van der Waals et augmenter la stabilité colloïdale. Par exemple, le choix du solvant possédant l’indice de réfraction le plus proche possible de celui des particules peut diminuer les forces de van der Waals. D'autres facteurs influencent la stabilité comme les interactions électrostatiques et les interactions spécifiques liées aux chaînes de polymères. Dans le cas des polymères, les chaînes peuvent être greffées à la surface des particules ou être dissoutes dans le solvant (chaînes libres). Dans ce travail de thèse, nous avons étudié l’effet de la stabilisation par déplétion dans le cas des chaînes de polymères libres (FPI, "free polymer induced interaction"). Des modèles théoriques précédents portent un caractère trop simplifié et utilisent des approximations sans vérification. De plus, l’influence des paramètres de la solution, c’est-à-dire, de la structure de polymères et de son interaction avec la surface de particule, n’a pas été étudiée.Les chaînes polymères libres ont été modélisées comme des marches aléatoires dans un champ moléculaire auto-cohérent qui satisfait à l'équation intégro-différentielle de diffusion. Pour le domaine moléculaire, nous avons utilisé un potentiel chimique qui, pour la solution de polymère semi-dilué, peut être représenté comme une expansion du viriel où nous n’avons pris en considération que les deuxième et troisième coefficients du viriel de la solution de polymère. En variant des paramètres tels que la rigidité du polymère, la longueur du polymère, la concentration en polymère et le régime du solvant (comme le solvant thêta), que ce soit pour une surface colloïdale purement répulsive, pour une surface adsorbée ou pour la surface d'une couche de polymère greffé, nous avons été en mesure d'améliorer la barrière répulsive due aux polymères libres entre les particules et donc nous avons trouvé des conditions de la stabilisation cinétique du système. / Stable colloidal dispersions with evenly distributed particles are important for many technological applications. Due to Brownian motion colloidal particles have constant collisions with each other which often lead to their aggregation driven by the long range van der Waals attraction. As a result the colloidal systems often tend to precipitate. A number of methods have been devised to minimize the effect of long-range van der Waals attraction between colloidal particles or to override the influence of the attraction in order to provide the colloidal stability.In the PhD thesis we investigated the colloidal stabilization in solutions of free polymers which is commonly referred to as depletion stabilization. Previous theoretical studies of free-polymer induced (FPI) stabilization were based on oversimplified models involving uncontrolled approximations. Even the most basic features of the depletion stabilization phenomenon were unknown. It was unclear how the PI repulsion depends on the solution parameters, polymer structure and monomer/surface interactions.The free polymer chains were modeled as random walks in a self-consistent molecular field that satisfied to diffusion-like integro-differential equation. As the molecular field we used the chemical potential that for semi-dilute polymer solution can be represented as a virial expansion where we took into account only second and third virial coefficients of the polymer solution. Varying the parameters like polymer stiffness, polymer length, polymer concentration and solvent regime (like theta solvent) whether it is for purely repulsive colloidal surface, adsorbed surface or surface with grafted polymer layer we were able to enhance the repulsive barrier due to the free polymers between the particles and therefore found conditions for kinetic stabilization of the system.

>Magister Scientiae - MSc / This research undertakes to investigate both long-term and short-term investor overreaction on the JSE Limited (JSE) over the period from 1 January 2002 to 31 December 2009. The period covers the restructuring and reform of the JSE in the early 2000s to the end of global financial market crisis in late 2008/2009, which can be regarded as a complete economic cycle. The performances of the winner and loser portfolios are evaluated by assessing their cumulative abnormal returns (CAR) over a 24-month holding period. The test results show no evidence of mean reversion for winner and loser portfolios formed based on prior returns of 12 months or less. However, test results show evidence of significant mean reversion for the winner and loser portfolios constructed based on their prior 24 months and 36 months returns. In addition, the study reveals that the mean reversion is more significant for longer-formation-period portfolios as well as for longer holding periods. The examination of the cumulative loser-winner spreads obtained from the contrarian portfolios based on the constituents’ prior 24 month and 36 month returns indicates that the contrarian returns increase for portfolios formed between 2004 and 2006, and declines thereafter towards the end of the examination period. The deterioration of contrarian returns coincides with the subprime mortgage crisis in 2007 and the subsequent global financial crisis in 2008. This evidence suggests that the degree of mean reversion on the JSE is positively correlated to the South African business cycle.

Nous étudions le modèle de la marche aléatoire branchante. Nous obtenons d'abord des résultats concernant le processus ponctuel formé par les particules extrémales, résolvant ainsi une conjecture de Brunet et Derrida 2010 [36]. Ensuite, nous établissons la dérivée au point critique de la limite des martingales additives complétant ainsi l'étude initiée par Biggins [23]. Ces deux travaux reposent sur les techniques modernes de décompositions épinales de la marche aléatoire branchante, originairement développées par Chauvin, Rouault et Wakolbinger [41], Lyons, Pemantle et Peres [74], Lyons [73] et Biggins et Kyprianou [24]. Le dernier chapitre de la thèse porte sur un champ Gaussien log-correle introduit par Kahane 1985 [61]. Via de récents travaux comme ceux de Allez, Rhodes et Vargas [11], Duplantier, Rhodes, Sheeld et Vargas [46] [47], ce modèle a connu un important regain d'intérêt. La construction du chaos multiplicatif Gaussien dans le cas critique a notamment été prouvée dans [46]. S'inspirant des techniques utilisées pour la marche aléatoire branchante nous résolvons une conjecture de [46] concernant le maximum de ce champ Gaussien. / We study the model of the branching random walk. First we obtain some results concerning thepoint process formed by the extremal particles, proving a Brunet and Derrida's conjecture [36] as well. Thenwe establish the derivative of the additive martingale limit at the critical point, completing the study initiatedby Biggins [23]. These two works rely on the spinal decomposition of the branching random walk, originallyintroduced by Chauvin, Rouault and Wakolbinger [41], Lyons, Pemantle and Peres [74], Lyons [73] and Bigginsand Kyprianou [24].The last chapter of the thesis deals with a log-correlated Gaussian field introduced by Kahane [61]. Thismodel was recently revived in particular by Allez, Rhodes and Vargas [11], and Duplantier, Rhodes, Shefield andVargas [46] [47]. Inspired by the techniques used for branching random walk we solved a conjecture of Duplantier,Rhodes, Shefield and Vargas [46], on the maximum of this Gaussian field.

The Johannesburg Stock Exchange market was tested for the existence of the random walk hypothesis using All Share Index (ALSI) and time series data for the period between 2000 and 2011. The traditionally used methods, the unit root tests and autocorrelation test were employed first and they all confirmed that during the period under consideration, the JSE price index followed the random walk process. In addition, the ARIMA model was built and it was found that the ARIMA ( 1, 1, 1) was the model that best fitted the data in question. Furthermore, residual tests to help determine whether the residuals of the estimated equation show random walk process in the series were done. It was found that the ALSI resembles series that follow random walk hypothesis with strong evidence of RWH indicated in the conducted forecasting tests which showed vast variance between forecasted values and actual indicating little or no forecasting strength in the series. To further validate the findings in this research, the variance ratio test was conducted under heteroscedasticity and it also strongly corroborated that the existence of a random walk process cannot be rejected in the JSE. It was concluded that since the returns follow the random walk hypothesis, it can be said that JSE is efficient in the weak form level of the EMH and therefore opportunities of making excess returns based on out- performing the market is ruled out and is merely a game of chance. In other words, it will be of no use to choose stocks based on information about recent trends in stock prices.

In the stochastic modeling of neurons, the first passage time problem arises as a natural object of study when considering the inter spike interval distribution. In this report, we study some aspects of this problem as it arises in the context of neuroscience. In the first chapter we describe the basic neurophysiology required to model the neuron. In the second, we study the Poisson model, Stein’s model, and some diffusion models, calculating or indicating methods to compute the density of the first passage time random variable or its moments. In the third and fourth chapters, we study the Fokker-Planck equation, and use it to compute the first passage time in the discrete and continuous time random walk cases. In the final chapter, we study sequences of neurons and the change in the density of the waiting time distributions, and hence in the inter spike intervals, as the output spike train from one neuron is considered as the input in the subsequent neuron.

Cette thèse étudie la moyennabilité et la propriété de Liouville des groupes pleins-topologiques des systèmes de Cantor, des groupes d'échanges d'intervalles, et des groupes agissants sur les arbres enracinés. Dans le Chapitre 2, nous obtenons les premiers exemples de groupes simples, infinis, de type fini, tels que le bord de Poisson de toute marche aléatoire simple est trivial (la propriété de Liouville). Ces exemples sont des sous-groupes dérivés de groupes pleins topologiques d'une famille de sous-décalages minimaux. Nous montrons que si la complexité d'un sous-décalage (pas nécessairement minimal) est strictement sous-quadratique, toute mesure de probabilité symétrique de support fini sur le groupe plein-topologique est d'entropie asymptotique nulle. Dans le Chapitre 3, nous exhibons une famille de groupes pleins-topologiques de sous-décalages minimaux qui contiennent les groupes de Grigorchuk G_ω comme sous-groupes. Cette construction montre que le groupe plein-topologique d'un sous-décalage minimal peut avoir des sous-groupes de croissance intermédiaire, en répondant à une question de Grigorchuk. Dans le Chapitre 4 (basé sur un travail en commun avec K. Juschenko, N. Monod, M. de la Salle) nous étudions les actions extensivement moyennables, une notion qui est un outil pour montrer la moyennabilité des groupes. Comme application, nous montrons la moyennabilité des groupes d'échanges d'intervalles dont les angles de translations ont rang rationnel au plus 2. Nous obtenons aussi une caractérisation "de type Kesten" de la moyennabilité extensive d'une action, et nous l'utilisons pour donner une preuve courte, purement probabiliste du fait que les actions récurrentes sont extensivement moyennables. Nous étudions aussi la propriété de Liouville pour les groupes d'échanges d'intervalles, et nous montrons qu'il existe des groupes d'échanges d'intervalles tels que toute mesure de support fini non dégénérée a un bord non trivial. Dans le Chapitre 5 (basé sur un travail en commun avec G. Amir, O. Angel, B. Virág) nous montrons que les groupes agissant sur les arbres enracinés par automorphismes bornés ont la propriété de Liouville. En particulier cela inclut les groupes engendrés par des automates d'activité bornée. / This thesis deals with the Liouville property and amenability of topological full groups of Cantor systems, groups of interval exchanges, and groups acting on rooted trees. In Chapter 2, we provide the first examples of finitely generated, infinite simple groups that have trivial Poisson-Furstenberg boundary for simple random walks (the Liouville property). These arise as the derived subgroup of the topological full groups of a family of minimal subshifts. We show that if the complexity of a (non necessarily minimal) subshift grows strictly subquadratically, every symmetric and finitely supported probability measure on the topological full group has vanishing asymptotic entropy. In Chapter 3, we exhibit a family of topological full groups of minimal subshifts that contain Grigorchuk groups G_ω as subgroups. This shows that the topological full group of a minimal subshift can have subgroups of intermediate growth, answering a question of Grigorchuk. In Chapter 4 (based on a joint work with K. Juschenko, N. Monod, M. de la Salle), we study various features of extensively amenable group actions, a notion which is a tool to prove amenability of groups. As an application, we prove amenability of groups of interval exchanges whose angular components have rational rank at most 2. We also obtain a "Kesten-like" characterisation of extensive amenability in terms of the inverted orbit and use it give a short, probabilistic proof of the fact that recurrent actions are extensively amenable. Finally we study the Liouville property for groups of interval exchanges, and show that there are groups of interval exchanges that admit no finitely supported measure with trivial boundary. In Chapter 5 (based on a joint work with G. Amir, O. Angel, B. Virág), we establish the Liouville property for all groups acting on rooted trees by bounded automorphisms. This includes in particular groups generated by bounded automata. This strengthens results by various authors about amenability of these groups, some of which are based on proving the Liouville property in some special cases.