This thesis consists of two papers devoted to the asymptotics of random matrix ensembles and measure valued stochastic processes which can be considered as generalizations of the Gaussian unitary ensemble (GUE) of Hermitian matrices H=A+A†, where the entries of A are independent identically distributed (iid) centered complex Gaussian random variables. In the first paper, a system of interacting diffusing particles on the real line is studied; special cases include the eigenvalue dynamics of matrix-valued Ornstein-Uhlenbeck processes (Dyson's Brownian motion). It is known that the empirical measure process converges weakly to a deterministic measure-valued function and that the appropriately rescaled fluctuations around this limit converge weakly to a Gaussian distribution-valued process. For a large class of analytic test functions, explicit formulae are derived for the mean and covariance functionals of this fluctuation process. The second paper concerns a family of random matrix ensembles interpolating between the GUE and the Ginibre ensemble of n x n matrices with iid centered complex Gaussian entries. The asymptotic spectral distribution in these models is uniform in an ellipse in the complex plane, which collapses to an interval of the real line as the degree of non-Hermiticity diminishes. Scaling limit theorems are proven for the eigenvalue point process at the rightmost edge of the spectrum, and it is shown that a non-trivial transition occurs between Poisson and Airy point process statistics when the ratio of the axes of the supporting ellipse is of order n -1/3. / Denna avhandling består av två vetenskapliga artiklar som handlar om gränsvärdessatser för slumpmatriser och måttvärda stokastiska processer. De modeller som studeras kan betraktas som generaliseringar av den gaussiska unitära ensembeln (GUE) av hermiteska n x n-matriser H=A+A†, där A är en matris vars element är oberoende, likafördelade, centrerade, komplexa normalfördelade stokastiska variabler. I artikel I betraktas ett system av växelverkande diffunderande partiklar på reella linjen, vissa specialfall av denna modell kan tolkas som egenvärdesdynamiken för matrisvärda Ornstein-Uhlenbeck-processer (Dysons brownska rörelse). Sedan tidigare är det känt att den empiriska måttprocessen konvergerar svagt mot en deterministisk måttvärd funktion och att fluktuationerna runt denna gräns, i lämplig skalning, konvergerer svagt mot en distributionsvärd gaussisk process. För en stor klass av analytiska testfunktioner härleds explicita formler för medelvärdes- och kovariansfunktionalerna för denna fluktuationsprocess. Artikel II behandlar en familj av slumpmatrisensembler som interpolerar mellan GUE och Ginibre-ensembeln, bestående av matriser A som ovan. För denna modell är egenvärdena komplexa och asymptotiskt likformigt fördelade i en ellips i komplexa planet. Skalningsgränsvärdessatser för egenvärdet med maximal realdel och för egenvärdespunktprocessen kring detta visas för ett allmänt val av interpolationsparametern i modellen. Då förhållandet mellan axlarna i den asymptotiska ellipsen är av storleksordning n-1/3 uppträder en övergångsfas mellan Airypunktprocess- och Poissonprocessbeteendena, typiska för GUE respektive Ginibre-ensembeln. / QC 20100705

Šiame magistriniame darbe yra nagrinėjami nepriklausomi vienodai pasiskirstę atsitiktiniai dydžiai, turintys visus absoliutinius baigtinius momentus. Magistrinio darbo tikslas - atlikti konvergavimo greičio į normalųjį dėsnį įvertinimą. Darbą sudaro aštuoni skyriai. Įvade aprašoma problema ir visi tyrimo parametrai. Antrasis skyrius skirtas teoriniai analizei. Šiame skyriuje pateikiamos svarbiausios teorinės žinios ir metodai, kurie bus taikomi magistrinio darbo uždaviniams bei tikslams įgyvendinti. Trečiame skyriuje nagrinėjami kumuliantai Bernulio schemos atveju, o ketvirtame - analizuojamas Čebyšovo asimptotinis skleidinys ir pasinaudojus matematiniu paketu Maple, grafiniu būdu, tyrinėjamas jo konvergavimas. Aproksimacijos normaliuoju dėsniu tikslumui įvertinti naudojamas charakteristinių funkcijų metodas, todėl penktasis skyrius yra skiriamas suglodinimo nelygybių patikslinimui. Šeštame skyriuje, pasinaudojus turimais rezultatais, realizuojamas magistrinio darbo tikslas, o septintame - patikrinamas absoliutinės paklaidos įvertis Bernulio schemos atveju. Išvados ir rezultatai glaustai išdėstomi aštuntame skyriuje. / This master thesis considers independiant and identically distributed random variables, having absolute finite moments. The main task is to determine error estimate of the normal approximation. The work consists of eight chapters. In the introduction are considered problems and all subjects of research. The second chapter is designed for the theory analysis. Here are placed the main theoretical studies and methods that are used to solve the aims of the master thesis. The third chapter is intended to deal with cumulants in case of the Bernoulli’s distribution, the fourth one - is analyzing the Čebyšova’s asymptotic expansion and it convergence with the help of the mathematical package Maple. The method of characteristic’s functions is used to find the remainder term of the normal approximation, so the fifth chapter is designed to specify smoothing inequalities. Based on these results, the main task of the master thesis was obtained and specified in the sixth chapter. In the seventh one the error estimate in case of Bernoulli’s distribution, was examined with a mathematical package Maple. The short conclusions and results are placed in the eighth chapter.

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Three classes of structural change tests (or tests for parameter instability) which have been receiving much attention in both the statistics and econometrics communities but have been developed in rather loosely connected lines of research are unified by embedding them into the framework of generalized M-fluctuation tests (Zeileis and Hornik, 2003). These classes are tests based on F statistics (supF, aveF, expF tests), on OLS residuals (OLS-based CUSUM and MOSUM tests) and on maximum likelihood scores (including the Nyblom-Hansen test). We show that (represantives from) these classes are special cases of the generalized M-fluctuation tests, based on the same functional central limit theorem, but employing different functionals for capturing excessive fluctuations. After embedding these tests into the same framework and thus understanding the relationship between these procedures for testing in historical samples, it is shown how the tests can also be extended to a monitoring situation. This is achieved by establishing a general M-fluctuation monitoring procedure and then applying the different functionals corresponding to monitoring with F statistics, OLS residuals and ML scores. In particular, an extension of the supF test to a monitoring scenario is suggested and illustrated on a real-world data set. / Series: Research Report Series / Department of Statistics and Mathematics

Dans cette thèse nous étudions les théorèmes limites dans l’analyse statistique dessystèmes dynamiques. Le premier chapitre est consacré aux notions des bases des systèmesdynamiques ainsi que la théorie ergodique. Dans le deuxième chapitre nous introduisonsun cadre fonctionnel abstrait pour lequel la version quenched du théorème de la limitecentrale (TLC) en dimension 1 pour les systèmes dynamiques uniformément dilatantsest satisfaite sous une condition de validité nécessaire et suffisante. Le troisième chapitreest consacré au principe d’invariance presque sûr (PIPS) pour les application aléatoiresdilatantes par morceaux. Nous présentons certaines hypothèses sous lesquelles le (PIPS)est vérifié en utilisant la méthode d’approximation des martingales de Cuny et Merlèvede.Nous étudions aussi le théorème de Sprindzuk et ses conséquences. Nous établissons dansle chapitre quatre la décroissance des corrélations pour les systèmes dynamiques aléatoiresuniformément dilatants par la méthode de couplage en dimension 1. Nous terminons cetravail par une présentation des concepts de base de la théorie des mesures et probabilitéset une présentation de l’espace des fonctions à variation bornée. / In this thesis we study the limit theorems in the statistical analysis of dynamicalsystems. The first chapter is devoted to the basic notions in dynamical systems as well asthe ergodic theory. In the second chapter we introduce an abstract functional frameworkunder which the quenched version of the central limit theorem (CLT) in dimension 1for uniformly expanding dynamic systems is satisfied under a necessary and sufficientcondition validity. The third chapter is devoted to the almost sure invariance principle(ASIP) for random piecewise expanding maps. We present some hypotheses under whichthe (ASIP) is verified using the method of approximation of the martingales of Cuny andMerlèvede. We also study the Sprindzuk theorem and its consequences. In chapter four,we define the decay of correlations for the random dynamical systems uniformly expandingby the coupling method in dimension 1. We finish this work with a presentation of thebasic concepts of the theory of measures and probabilities and a presentation of the spaceof functions with bounded variation.

"The purposes of this study were: (1) to investigate the effectiveness of the computer-student interactive method in presenting statistical concepts and in instructing students in the applications of these concepts, and (2) to develop instruments that test for the understanding of these concepts and the mastery of these application skills"--Abstract. / Typescript. / "Spring Semester, 1990." / "Submitted to the Department of Curriculum and Instruction in partial fulfillment of the requirements for the degree of Doctor of Philosophy." / Advisor: E. T. Denmark, Professor Directing Dissertation. / Includes bibliographical references.

Made available in DSpace on 2016-04-27T16:59:00Z (GMT). No. of bitstreams: 1 Chang Kuo Rodrigues.pdf: 19165521 bytes, checksum: 423ed2c3982a3973f316dec156e2d596 (MD5) Previous issue date: 2009-12-02 / This paper refers to the building of mathematical and/or statistical ideas and concepts around Central Limit Theorem for Mathematics graduates.The investigation focuses the importance of the theorem in Statistics Inference and its comprehension by the professionals to be, who will act in Basic Education. Therefore, we chose to research some books related to the teaching and learning process of the theorem and emphasised its importance on the Mathematics teacher daily practice. The theoretical approach is about Mathematics Teaching theories, particularly the Theory of Didactic Transposition ( CHEVALLARD, 1985), with an echological approach under the knowlwdge and teaching point of view ( ARTAUD, 1998). We chose methodological procedures directed to the didactic design (ARTIGUE, 2009), with qualitative nature, and whose assumptions are linked to Teaching Engineering (ARTIGUE, 1988). The subjects of this investigation are the graduates who had some knowledge about Basic Statistics and, from a previous analysis about the kind of knowledge they had about the theme, we presented some activities in a problem-situation context connected to the Mathematics teachers daily practice. The analysis of these results allowed us to relate the existing problems between the subject and the students from Basic Education, which involved statistics literacy. After these activities, there was a dialogue, with discussions about the theme, allowing us to analyse how the ideas and concepts around the Central Limit Theorem were built, being its comprehension the main aim for the graduates. Besides that, we analysed some textbooks for higher education, based on the Anthropological Theory of Didactic (CHEVALLARD, 1996, 1999), which also showed us the essential knowledge for the theorem to live , because the approach is under the knowledge and teaching echological point of view. On the other hand, we detected what kind of limitations, or restrictions, exist in the books analysed, interfering in the elaboration of the activities by the teacher. Thus, our investigation reaffirms the importance of teaching and learning Statistics in the various applications for the Mathematics teachers to be formation in a world controlled by the technological advances, which interfere directly on the understanding of the information we receive every moment / O presente trabalho refere-se à construção das ideias e dos conceitos matemáticos e/ou estatísticos em torno do Teorema Central do Limite para os Licenciandos de Matemática. O cerne da investigação limita-se à importância do teorema na Inferência Estatística e à sua compreensão pelos futuros profissionais que atuarão na Educação Básica. Nesse sentido, optamos por revisar algumas bibliografias que têm relação com o processo de ensino e de aprendizagem do teorema e enfatizamos sua importância na pratica do dia a dia do professor de Matemática. O quadro teórico incide sobre as teorias da Didática da Matemática, particularmente, a Teoria da Transposição Didática (CHEVALLARD, 1985), munido de uma abordagem ecológica sob o ponto de vista do saber e do didático (ARTAUD, 1998). Optamos por procedimentos metodológicos voltados para o design didático (ARTIGUE, 2009), de cunho qualitativo e, cujos pressupostos estão aliados à Engenharia Didática (ARTIGUE, 1988). Os sujeitos dessa investigação são os licenciandos que já predispunham de conhecimentos sobre a Estatística Básica e, a partir de uma análise prévia sobre que tipos de conhecimento eles já detinham sobre o tema, apresentamos algumas atividades no contexto de uma situação-problema pertinente ao cotidiano dos professores de Matemática. A análise desses resultados nos propiciou interrelacionar as problemáticas existentes na disciplina de Matemática com alunos da Educação Básica, envolvendo assim, a literacia estatística. Após a realização dessas atividades, ocorreu também um diálogo, com discussões acerca do tema, o que nos permitiu analisar como foram construídos as ideias e os conceitos no entorno do Teorema Central do Limite, de modo que sua compreensão fosse o principal alvo para os licenciandos. Além disso, analisamos alguns livrostexto do ensino superior, à luz da Teoria Antropológica do Didático (CHEVALLARD, 1996, 1999), o que também nos indicou que saberes são indispensáveis de modo que o teorema viva , já que a abordagem é sob o ponto de vista ecológico do saber e do didático. Por outro lado, detectamos que tipos de limitações, ou restrições, existem nas obras consultadas, interferindo assim, a elaboração das atividades por parte do professor. Portanto, a nossa investigação reitera a importância do ensino e da aprendizagem da Estatística nas diversas aplicações na formação dos futuros professores de Matemática num mundo ditado pelos avanços tecnológicos, que interferem diretamente na leitura de informações que recebemos a todo instante

This thesis deals with modeling of particle processes. In the first part we ex- amine Gibbs facet process on a bounded window with discrete orientation distri- bution and we derive central limit theorem (CLT) for U-statistics of facet process with increasing intensity. We calculate all asymptotic joint moments for interac- tion U-statistics and use the method of moments for deriving the CLT. Moreover we present an alternative proof which makes use of the CLT for U-statistics of a Poisson facet process. In the second part we model planar segment processes given by a density with respect to the Poisson process. Parametric models involve reference distributions of directions and/or lengths of segments. Statistical methods are presented which first estimate scalar parameters by known approaches and then the reference distribution is estimated non-parametrically. We also introduce the Takacs-Fiksel estimate and demonstrate the use of estimators in a simulation study and also using data from actin fibres from stem cells images. In the third part we study a stationary Gibbs particle process with determin- istically bounded particles on Euclidean space defined in terms of a finite range potential and an activity parameter. For small activity parameters, we prove the CLT for certain statistics of this...

<p>This thesis consists of four papers on the following topics in analysis and probability: analysis on Wiener space, asymptotic properties of orthogonal polynomials, and convergence rates in the central limit theorem. The first paper gives lower bounds on the constants in the Meyer inequality from the Malliavin calculus. It is shown that both constants grow at least like <i>(p-1)</i>^-1 or like <i>p</i> when <i>p</i> approaches 1 or ∞ respectively. This agrees with known upper bounds. In the second paper, an extremal problem on Wiener chaos motivates an investigation of the <i>L</i>^p-norms of Hermite polynomials. This is followed up by similar computations for Charlier polynomials in the third paper. In both cases, the <i>L</i>^p-norms present a peculiar behaviour with certain threshold values of p, where the growth rate and the dominating intervals undergo a rapid change. The fourth paper analyzes a connection between probability and numerical analysis. More precisely, known estimates on the convergence rate of finite difference equations are "translated" into results on convergence rates of certain functionals in the central limit theorem. These are also extended, using interpolation of Banach spaces as a main tool. Besov spaces play a central role in the emerging results.</p>