This PhD thesis consists of a summary and five papers which all deal with asymptotic problems on certain homogeneous spaces. In Paper I we prove asymptotic equidistribution results for pieces of large closed horospheres in cofinite hyperbolic manifolds of arbitrary dimension. All our results are given with precise estimates on the rates of convergence to equidistribution. Papers II and III are concerned with statistical problems on the space of n-dimensional lattices of covolume one. In Paper II we study the distribution of lengths of non-zero lattice vectors in a random lattice of large dimension. We prove that these lengths, when properly normalized, determine a stochastic process that, as the dimension n tends to infinity, converges weakly to a Poisson process on the positive real line with intensity 1/2. In Paper III we complement this result by proving that the asymptotic distribution of the angles between the shortest non-zero vectors in a random lattice is that of a family of independent Gaussians. In Papers IV and V we investigate the value distribution of the Epstein zeta function along the real axis. In Paper IV we determine the asymptotic value distribution and moments of the Epstein zeta function to the right of the critical strip as the dimension of the underlying space of lattices tends to infinity. In Paper V we determine the asymptotic value distribution of the Epstein zeta function also in the critical strip. As a special case we deduce a result on the asymptotic value distribution of the height function for flat tori. Furthermore, applying our results we discuss a question posed by Sarnak and Strömbergsson as to whether there in large dimensions exist lattices for which the Epstein zeta function has no zeros on the positive real line.

The study of properties of probability distributions has always been a persistent theme of statistics and of applied probability. This thesis deals with an investigation of distribution functions under the following two topics: (i) characterization of distributions based on record values and order statistics, (ii) properties of the skew-t distribution. Within the extensive characterization literature there are several results involving properties of record values and order statistics. Although there have been many well known results already developed, it is still of great interest to find new characterization of distributions based on record values and order statistics. In the first part, we provide the conditional distribution of any record value given the maximum order statistics and study characterizations of distributions based on record values and the maximum order statistics. We also give some characterizations of the mean value function within the class of order statistics point processes, by using certain relations between the conditional moments of the jump times or current lives. These results can be applied to characterize the uniform distribution using the sequence of order statistics, and the exponential distribution using the sequence of record values, respectively. Azzalini (1985, 1986) introduced the skew-normal distribution which includes the normal distribution and has some properties like the normal and yet is skew. This class of distributions is useful in studying robustness and for modeling skewness. Since then, skew-symmetric distributions have been proposed by many authors. In the second part, the so-called generalized skew-t distribution is defined and studied. Examples of distributions in this class, generated by the ratio of two independent skew-symmetric distributions, are given. We also investigate properties of the skew-symmetric distribution.

This paper deals with CUSUM tests for monitoring of intensity parameter of a Poisson process if this process can be observed in a restricted manner only at pregiven equidistant time points. In this case the process can be monitored by means of a CUSUM test for the parameter of a corresponding Poisson distribution. For rational reference parameter values the computation of average run length is reduced to that of solving of a system of simultaneous linear equations. The performance of obtained CUSUM tests is discussed by means of corresponding examples.

Baigiamajame magistro darbe gautos dvejetainių atstatymo procesų sumų konvergavimo į Puasono procesą sąlygos. Remiamasi Grigelionio teorema, nusakančia nepriklausomų taškinių procesų sumų konvergavimo sąlygas. Analizuojami atvejai, kai sumuojamų dvejetainių atstatymo procesų veikimo ir atstatymo periodai yra nepriklausomi ir pasiskirstę pagal tolygųjį, eksponentinį, geometrinį ir Erlango dėsnius. Taip pat nagrinėjamas atvejis, kai veikimo ir atstatymo laikotarpiai turi skirtingus skirstinius. Kiekvienu atveju gautos ir įrodytos būtinos ir pakankamos sąlygos. Remiantis teoriniais rezultatais, procesai yra modeliuojami ir lyginami. Darbo pabaigoje yra suformuluojamos išvados. / In this master thesis conditions for convergence of sums of alternating renewal processes to Poisson process is obtained. Thesis is based on Grigelionis theorem, which defines conditions for convergence of sums of independent counting processes. More specific cases, when alternating renewal processes life and recovery periods are independent and have uniform, exponential, geometric and Erlang distributions, are examined too. Also, case when life and recovery periods have different distributions is examined. Necessary and sufficient conditions are formulated and proven for each case. Processes are modeled and compared according to theoretical results. In the end of thesis conclusions are made.

Interessado em fazer com que o seu capital gere lucros, o investidor ao optar por negociar ativos, fica sujeito aos riscos econômicos de qualquer negociação, pois não existe uma certeza quanto a valorização ou desvalorização de um ativo. Eis que surge o mercado futuro, em que é possível negociar contratos a fim de se proteger (hedge) dos riscos de perdas ou ganhos excessivos, fazendo com que a compra ou venda de ativos, seja justa para ambas as partes. O objetivo deste trabalho consiste em estudar os processos de Lévy de puro salto de atividade finita, também conhecido como modelo de Poisson composto, e suas aplicações. Proposto pelo matemático francês Paul Pierre Lévy, os processos de Lévy tem como principal característica admitir saltos em sua trajetória, o que é frequentemente observado no mercado financeiro. Determinaremos uma estratégia de hedging no modelo de mercado com o processo de Poisson composto via o conceito de mean-variance hedging e princípio da programação dinâmica. / The investor, that negotiate assets, is subject to economic risks of any negotiation because there is no certainty regarding the appreciation or depreciation of an asset. Here comes the futures market, where contracts can be negotiated in order to protect (hedge) the risk of excessive losses or gains, making the purchase or sale assets, fair for both sides. The goal of this work consist in study Lévy pure-jump process with finite activity, also known as compound Poisson process, and its applications. Discovered by the French mathematician Paul Pierre Lévy, the Lévy processes admits jumps in paths, which is often observed in financial markets. We will define a hedging strategy for a market model with compound Poisson process using mean-variance hedging and dynamic programming.

Submitted by Caroline Periotto (carol@ufscar.br) on 2016-09-09T19:56:20Z No. of bitstreams: 1 DissVSHS.pdf: 882234 bytes, checksum: f08aea79440ba666e616318257bbdec9 (MD5) / Approved for entry into archive by Marina Freitas (marinapf@ufscar.br) on 2016-09-13T14:05:42Z (GMT) No. of bitstreams: 1 DissVSHS.pdf: 882234 bytes, checksum: f08aea79440ba666e616318257bbdec9 (MD5) / Approved for entry into archive by Marina Freitas (marinapf@ufscar.br) on 2016-09-13T14:05:49Z (GMT) No. of bitstreams: 1 DissVSHS.pdf: 882234 bytes, checksum: f08aea79440ba666e616318257bbdec9 (MD5) / Made available in DSpace on 2016-09-13T14:05:56Z (GMT). No. of bitstreams: 1 DissVSHS.pdf: 882234 bytes, checksum: f08aea79440ba666e616318257bbdec9 (MD5) Previous issue date: 2015-12-07 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / The investor, that negotiate assets, is subject to economic risks of any negotiation because there is no certainty regarding the appreciation or depreciation of an asset. Here comes the futures market, where contracts can be negotiated in order to protect (hedge) the risk of excessive losses or gains, making the purchase or sale assets, fair for both sides. The goal of this work consist in study Lévy pure-jump process with finite activity, also known as compound Poisson process, and its applications. Discovered by the French mathematician Paul Pierre Lévy, the Lévy processes admits jumps in paths, which is often observed in financial markets. We will define a hedging strategy for a market model with compound Poisson process using mean-variance hedging and dynamic programming. / Interessado em fazer com que o seu capital gere lucros, o investidor ao optar por negociar ativos, fica sujeito aos riscos econômicos de qualquer negociação, pois não existe uma certeza quanto a valorização ou desvalorização de um ativo. Eis que surge o mercado futuro, em que é possível negociar contratos a fim de se proteger (hedge) dos riscos de perdas ou ganhos excessivos, fazendo com que a compra ou venda de ativos, seja justa para ambas as partes. O objetivo deste trabalho consiste em estudar os processos de Lévy de puro salto de atividade finita, também conhecido como modelo de Poisson composto, e suas aplicações. Proposto pelo matemático francês Paul Pierre Lévy, os processos de Lévy tem como principal característica admitir saltos em sua trajetória, o que é frequentemente observado no mercado financeiro. Determinaremos uma estratégia de hedging no modelo de mercado com o processo de Poisson composto via o conceito de mean-variance hedging e princípio da programação dinâmica.

Submitted by Izabel Franco (izabel-franco@ufscar.br) on 2016-09-21T19:45:33Z No. of bitstreams: 1 TeseMACM.pdf: 1006551 bytes, checksum: 49a419a1f18e05827f34631564fe431b (MD5) / Approved for entry into archive by Ronildo Prado (ronisp@ufscar.br) on 2016-09-28T18:19:58Z (GMT) No. of bitstreams: 1 TeseMACM.pdf: 1006551 bytes, checksum: 49a419a1f18e05827f34631564fe431b (MD5) / Approved for entry into archive by Ronildo Prado (ronisp@ufscar.br) on 2016-09-28T18:20:07Z (GMT) No. of bitstreams: 1 TeseMACM.pdf: 1006551 bytes, checksum: 49a419a1f18e05827f34631564fe431b (MD5) / Made available in DSpace on 2016-09-28T18:26:37Z (GMT). No. of bitstreams: 1 TeseMACM.pdf: 1006551 bytes, checksum: 49a419a1f18e05827f34631564fe431b (MD5) Previous issue date: 2015-09-02 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / In this thesis it is proposed alternative methodologies and extensions on models for recurrent event data. Speci cally, we propose a model in which the distribution of the gap time is easily derived from the marginal rate function providing more direct practical interpretation besides to consider the relation between successive gap times for each individual. Another model that extends the frailty models for recurrent event data to allow a Bernoulli, Geometric, Poisson, Discrete Weibull, Negative Binomial or other discrete distribution of the frailty variable has also been proposed. The parameter estimation procedure for both models was conducted considering maximum likelihood methods. Simulation studies were performed in order to examine some frequentist properties of the estimation method and evaluate the maximum likelihood estimates quality. Real data applications demonstrated the use of the proposed models. Overall, the proposed models were suitable for analyzing recurrent event data. / Neste trabalho propomos metodologias alternativas e extensões em modelos para dados de eventos recorrentes. Especificamente, propomos um modelo em que a distribuição condicional do tempo entre sucessivas ocorrências de um evento recorrente e derivada facilmente da função de taxa marginal, proporcionando interpretações praticas mais diretas, além de considerar a relação entre as sucessivas ocorrências para cada indivíduo. O outro modelo, que estende os modelos de fragilidade para dados de eventos recorrentes permitindo o uso de distribuições como Bernoulli, Geométrica, Poisson, Weibull Discreta, Binomial Negativa ou outra distribuição discreta para a variável de fragilidade, também foi proposto. O procedimento de estimação dos parâmetros para ambos modelos foi realizado considerando-se o método de máxima verossimilhança. Estudos de simulação foram realizados com o objetivo de analisar algumas propriedades frequentistas do método de estimação e avaliar a qualidade das estimativas de máxima verossimilhança. Aplicações a conjuntos de dados reais mostraram a aplicabilidade dos modelos propostos. De modo geral, os modelos propostos mostraram-se adequados para a modelagem de dados de eventos recorrentes.

Submitted by Franciele Moreira (francielemoreyra@gmail.com) on 2017-09-20T18:06:53Z No. of bitstreams: 2 Dissertação - Bruna Luiza de Faria Rezende- 2017.pdf: 1376216 bytes, checksum: 9c03a69f7f93de81123e21bc0a3a36da (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Approved for entry into archive by Luciana Ferreira (lucgeral@gmail.com) on 2017-09-21T10:52:16Z (GMT) No. of bitstreams: 2 Dissertação - Bruna Luiza de Faria Rezende- 2017.pdf: 1376216 bytes, checksum: 9c03a69f7f93de81123e21bc0a3a36da (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Made available in DSpace on 2017-09-21T10:52:16Z (GMT). No. of bitstreams: 2 Dissertação - Bruna Luiza de Faria Rezende- 2017.pdf: 1376216 bytes, checksum: 9c03a69f7f93de81123e21bc0a3a36da (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) Previous issue date: 2017-08-31 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / In this work a basic immigration process was investigated which starts with a single colony with a single individual at the origin of a homogeneous tree with the other empty vertices. The process colonies are established at the vertices of the graph and each one grows during a random time, according to a process of general counting until a disaster that annihilates part of the population occurs. After the collapse a random amount of individuals survives and attempts to establish, in a independent manner, new colonies in a neighboring vertices. After a time these formed colonies also suffer catastrophes and the process is repeated. It is important to emphasize that the time until the disaster of each colony is independent of the others. Here this general process was studied under two methods, Poisson growth with geometric catastrophe and Yule growth with binomial catastrophe. That is, in each colony the population grows following a Poisson (or Yule), process during a random time, considered here exponential, and soon after that time its size is reduced according to the geometric (or binomial) law. Conditions were analyzed in the set of parameters so that these processes survived and limits were established that were relevant for the probability of survival, the number of colonies generated during the process and the range of the colonies in relation to the initial point. / Neste trabalho foi investigado um processo básico de imigração o qual é iniciado com uma única colônia com um único indivíduo na origem de uma árvore homogênea com os demais vértices vazios. As colônias do processo se estabelecem nos vértices do grafo e cada uma cresce durante um tempo aleatório, de acordo com um processo de contagem geral até ocorrer um desastre que aniquila parte da população. Após o colapso uma quantidade aleatória de indivíduos sobrevive e tenta estabelecer, de forma independente, novas colônias em vértices vizinhos. Depois de um tempo essas colônias formadas também sofrem catástrofes e o processo se repete. É importante enfatizar que o tempo até o desastre de cada colônia independe do das demais. Aqui esse processo geral foi estudado sujeito a dois métodos, crescimento de Poisson com catástrofe geométrica e crescimento de Yule com catástrofe binomial. Ou seja, em cada colônia a população cresce seguindo um processo de Poisson (ou Yule), durante um tempo aleatório, considerado aqui exponencial, e logo após esse tempo seu tamanho é reduzido de acordo com a lei geométrica (ou binomial). Foram analisadas condições no conjunto de parâmetros para que esses processos sobrevivam e foram estabelecidos limites relevantes para a probabilidade de sobrevivência, o número de colônias geradas durante o processo e o alcance das colônias em relação ao ponto inicial.

Traffic modeling in computer networks has been researched for decades. A good model should reflect the features of real-world network traffic. With a good model, synthetic traffic data can be generated for experimental studies; network performance can be analysed mathematically; service provisioning and scheduling can be designed aligning with traffic changes. An important part of traffic modeling is to capture the dependence, either the dependence among different traffic flows or the temporal dependence within the same traffic flow. Nevertheless, the power of dependence models, especially those that capture the functional dependence, has not been fully explored in the domain of computer networks. This thesis studies copula theory, a theory to describe dependence between random variables, and applies it for better performance evaluation and network resource provisioning. We apply copula to model both contemporaneous dependence between traffic flows and temporal dependence within the same flow. The dependence models are powerful and capture the functional dependence beyond the linear scope. With numerical examples, real-world experiments and simulations, we show that copula modeling can benefit many applications in computer networks, including, for example, tightening performance bounds in statistical network calculus, capturing full dependence structure in Markov Modulated Poisson Process (MMPP), MMPP parameter estimation, and predictive resource provisioning for cloud-based composite services. / Graduate / 0984 / fdong@uvic.ca

Afin de garantir un MANET totalement connecté, nous proposons un outil analytique permettant d´estimer les rayons de couverture requis. Ensuite, nous étudions les communications multi-sauts et nous proposons plusieurs protocoles de routage pour améliorer les délais de communications. Puis, nous nous focalisons sur la contribution du réseau satellite en menant une étude comparative qualifiant les besoins en stations hybrides "HS" ainsi qu’une stratégie d´accès à ces passerelles satellite. Dans un deuxième temps, nous traitons l’ occurrence du dépassement des ressources radio et le dimensionnement de ces dernières de façon à optimiser la bande passante allouée au réseau. Nous dérivons un modèle analytique en utilisant des résultats issus de la géométrie aléatoire, permettant de prévoir la quantité de ressources radio requises par les noeuds actifs sous une certaine qualité de service "QoS" et plusieurs configurations d´antennes MIMO. Nous considérons tout d´abord un système d´accès centralisé où toutes les communications sont effectuées par l’ intermédiaire des noeuds chargés de la gestion des ressources radio. Ensuite, nous traitons le cas d´un système d´accès distribué sous le protocole d´accès Aloha où les nœuds sont autorisés à accéder à la bande passante partagée aléatoirement et uniformément. Les simulations et les résultats numériques permettent d´évaluer les performances en termes de bande passante requise, de capacité globale et de rayons de couverture. / In the first part of this thesis, we tackle the initialization of the network in this hierarchical context. We propose a statistical model enabling a network designer to perceive the requirements in terms of equipments, channel bandwidth, antenna configurations, antenna radiation pattern, achievable data rates for instance. In order to guarantee fully connected MANET, we introduce an analytical tool to estimate the required inter-staff-ships and inter-shipmasters coverage radii. Then we study the multi-hop end-to-end communications and we propose several routing protocols to enhance the delays. Afterwards, we focus on the contribution of the satellite backhaul with a comparative study qualifying the needs in Hybrid Stations “HSs” and a strategy to access to these gateways. In a second part, we emphasis on the radio resource outage occurrence and the dimensioning matter to optimize the allocated bandwidth to the network. We investigate stochastic geometry tools to provide an analytical model enabling to foresee the amount of required radio resources by the active nodes with a certain Quality of Service “QoS” and several Multiple Inputs Multiple Outputs “MIMO” antenna configurations in the maritime context. We consider first the centralized access scheme where all communications are performed via the shipmasters that are in charge of the radio resource management. Then we focus on the distributed access scheme with Aloha Medium Access Control “MAC” protocol where nodes are authorized to access to the shared bandwidth arbitrarily and unilaterally. Simulation and numerical results are provided to evaluate the performances in terms of required bandwidth, aggregate capacity.