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Alcântara, Daniel Tomás Vital de. "Central limit theorem variations." Master's thesis, Instituto Superior de Economia e Gestão, 2019. http://hdl.handle.net/10400.5/20409.
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Mestrado em Mathematical FinanceUm dos teoremas mais importantes da Teoria da Probabilidade é o Teorema do Limite Central. Este afirma que se Xn é uma sequência de variáveis aleatórias então as somas parciais normalizadas convergem para a distribuição normal. Além disso a ausência de pré condições faz-nos perguntar-nos se generalizações são possíveis. Particularmente neste manuscrito vamos focar-nos em duas questões: Existe uma taxa de convergência (universal) para o Teorema do Limite Central? Além disso em que circunstâncias podemos aplicar o Teorema do Limite Central? O teorema de Continuidade de Lévy afirma que a convergência em distribuição é equivalente à convergência nas funções características. Além disso quando aplicamos as expansões de Taylor a funções características ficamos com um polinómios com os momentos da variável como coeficientes. Por estas razões no nosso caso fazer os cálculos com funções características é preferível. Pelo teorema de Berry Essen podemos, de facto, encontrar a taxa de convergência que procuramos. E pelo teorema de Lindeberg e condição de Lyapunov podemos descobrir que o Teorema do Limite Central pode aplicar-se a sequências que não são identicamente distribuídas. Finalmente, utilizando o teorema ergódico vamos explicar como processos estocásticos estão relacionados com a teoria ergódica. Com isto vamos mostrar como este teorema pode ser utilizado pata encontrar um resultado quando a sequencia não é independente.
One of the most important theorems of Probability Theory is the Central Limit Theorem. It states that if Xn is a sequence of random variables then the normal- ized partial sums converge to a normal distribution. This result omits any rate of convergence. Furthermore the lack of assumptions makes us wonder if some gener- alizations are possible. Particularly in this essay we will focus on two questions: Does it exist a (uni- versal) rate of convergence for the Central Limit Theorem? Furthermore in which circumstances can we apply the Central Limit Theorem? The Lévy Continuity Theorem states that convergence on distribution functions is equivalent to convergence on characteristic functions. Furthermore when we ap- ply Taylor expansions to characteristic functions we get a polynomial with the mo- ments as coefficients. For these reasons, on our case computing with characteristic functions is preferable. By the Berry Essen Theorem we can in fact find the rate of convergence we are looking for. And by the Lindeberg Theorem and Lyapunov Condition we find that the Central Limit Theorem applies to sequences that are not identically distributed. Finally, using the Ergodic Theorem we will explain how stochastic processes are related to Ergodic Theory. With this we will show how this theorem can be used to find a result when the sequence is not independent.
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ALVES, RODRIGO BARRETO. "MARTINGALE CENTRAL LIMIT THEOREM." PONTIFÍCIA UNIVERSIDADE CATÓLICA DO RIO DE JANEIRO, 2017. http://www.maxwell.vrac.puc-rio.br/Busca_etds.php?strSecao=resultado&nrSeq=32327@1.
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PONTIFÍCIA UNIVERSIDADE CATÓLICA DO RIO DE JANEIROCOORDENAÇÃO DE APERFEIÇOAMENTO DO PESSOAL DE ENSINO SUPERIOR
PROGRAMA DE SUPORTE À PÓS-GRADUAÇÃO DE INSTS. DE ENSINO
PROGRAMA DE EXCELENCIA ACADEMICA
Esta dissertação é dedicada ao estudo das taxas de convergência no Teorema Central do Limite para Martingais. Começamos a primeira parte da tese apresentando a Teoria de Martingais, introduzindo o conceito de esperança condicional e suas propriedades. Desta forma poderemos descrever o que é um Martingal, mostraremos alguns exemplos, e exporemos alguns dos seus principais teoremas. Na segunda parte da tese vamos analisar o Teorema Central do Limite para variáveis aleatórias, apresentando os conceitos de função característica e convergência em distribuição, que serão utilizados nas provas de diferentes versões do Teorema Central do Limite. Demonstraremos três formas do Teorema Central do Limite, para variáveis aleatórias independentes e identicamente distribuídas, a de Lindeberg-Feller e para uma Poisson. Após, apresentaremos o Teorema Central do Limite para Martingais, demonstrando uma forma mais geral e depois enunciaremos uma forma mais específica a qual focaremos o resto da tese. Por fim iremos discutir as taxas de convergência no Teorema Central do Limite, com foco nas taxas de convergência no Teorema Central do Limite para Martingais. Em particular, exporemos o resultado de [4], o qual determina, até uma constante multiplicativa, a dependência ótima da taxa de um certo parâmetro do martingal.
This dissertation is devoted to the study of the rates of convergence in the Martingale Central Limit Theorem. We begin the first part presenting the Martingale Theory, introducing the concept of conditional expectation and its properties. In this way we can describe what a martingale is, present examples of martingales, and state some of the principal theorems and results about them. In the second part we will analyze the Central Limit Theorem for random variables, presenting the concepts of characteristic function and the convergence in distribution, which will be used in the proof of various versions of the Central Limit Theorem. We will demonstrate three different forms of the Central Limit Theorem, for independent and identically distributed random variables, Lindeberg-Feller and for a Poisson distribution. After that we can introduce the Martingale Central Limit Theorem, demonstrating a more general form and then stating a more specific form on which we shall focus. Lastly, we will discuss rates of convergence in the Central Limit Theorems, with a focus on the rates of convergence in the Martingale Central Limit Theorem. In particular, we state results of [4], which determine, up to a multiplicative constant, the optimal dependence of the rate on a certain parameter of the martingale.
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Sorokin, Yegor. "Probability theory, fourier transform and central limit theorem." Manhattan, Kan. : Kansas State University, 2009. http://hdl.handle.net/2097/1604.
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Jiang, Xinxin. "Central limit theorems for exchangeable random variables when limits are mixtures of normals /." Thesis, Connect to Dissertations & Theses @ Tufts University, 2001.
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Thesis (Ph.D.)--Tufts University, 2001.Adviser: Marjorie G. Hahn. Submitted to the Dept. of Mathematics. Includes bibliographical references (leaves44-46). Access restricted to members of the Tufts University community. Also available via the World Wide Web;
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Pramukkul, Pensri. "Temporal Complexity and Stochastic Central Limit Theorem." Thesis, University of North Texas, 2014. https://digital.library.unt.edu/ark:/67531/metadc700093/.
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Complex processes whose evolution in time rests on the occurrence of a large and random number of intermittent events are the systems under study. The mean time distance between two consecutive events is infinite, thereby violating the ergodic condition and activating at the same time a stochastic central limit theorem that explains why the Mittag-Leffler function is a universal property of nature. The time evolution of these complex systems is properly generated by means of fractional differential equations, thus leading to the interpretation of fractional trajectories as the average over many random trajectories, each of which fits the stochastic central limit theorem and the condition for the Mittag-Leffler universality. Additionally, the effect of noise on the generation of the Mittag-Leffler function is discussed. Fluctuations of relatively weak intensity can conceal the asymptotic inverse power law behavior of the Mittag-Leffler function, providing a reason why stretched exponentials are frequently found in nature. These results afford a more unified picture of complexity resting on the Mittag-Leffler function and encompassing the standard inverse power law definition.
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Humphreys, Natalia A. "A central limit theorem for complex-valued probabilities /." The Ohio State University, 1999. http://rave.ohiolink.edu/etdc/view?acc_num=osu1488187049540163.
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Zhang, Na. "Limit Theorems for Random Fields." University of Cincinnati / OhioLINK, 2019. http://rave.ohiolink.edu/etdc/view?acc_num=ucin1563527352284677.
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Mok, Kit Ying. "Central limit theorem for nonparametric regression under dependent data /." View Abstract or Full-Text, 2003. http://library.ust.hk/cgi/db/thesis.pl?MATH%202003%20MOK.
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Rahman, Mohammad Mahbubur. "Central Limit Theorem for some classes of dynamical systems." Thesis, National Library of Canada = Bibliothèque nationale du Canada, 1997. http://www.collectionscanada.ca/obj/s4/f2/dsk2/ftp04/mq25986.pdf.
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Thangavelu, Karthinathan. "Quantile estimation based on the almost sure central limit theorem." Doctoral thesis, [S.l.] : [s.n.], 2006. http://webdoc.sub.gwdg.de/diss/2006/thangavelu.
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Holzmann, Hajo. "Some remarks on the central limit theorem for stationary Markov processes." Doctoral thesis, [S.l.] : [s.n.], 2004. http://deposit.ddb.de/cgi-bin/dokserv?idn=972079874.
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Nakashima, Makoto. "Almost sure central limit theorem for branching random walks in random environment." 京都大学 (Kyoto University), 2012. http://hdl.handle.net/2433/157736.
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Paulsen, Michael Christoph. "Limit theorems for limit order books." Doctoral thesis, Humboldt-Universität zu Berlin, Mathematisch-Naturwissenschaftliche Fakultät II, 2014. http://dx.doi.org/10.18452/17023.
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Im ersten Teil der Dissertation wird ein diskretes stochastisches zustandsabhängiges Modell eines zweiseitigen Limit Orderbuchs als bestehend aus den Zustandsgrößen bester Bidpreis (Geldkurs), bester Askpreis (Briefkurs) und vorhandener Kauf- bzw. Verkaufsdichte definiert. Für eine einfache Skalierung mit zwei Zeitskalen wird ein Grenzwertsatz bewiesen. Die Veränderungen der besten Bid- und Askpreise werden im Sinne des Gesetzes der großen Zahlen skaliert und dies entspricht der langsameren Zeitskala. Das Platzieren bzw. Stornieren der Limitorder findet auf der schnelleren Zeitskala statt. Der Grenzwertsatz besagt, dass die fundamentalen Zustandsgrößen, gegeben Regularitätsbedingungen der einkommenden Order, fast sicher zu einem stetigen Limesmodell konvergieren. Im Limesmodell sind der beste Bidpreis und der beste Askpreis die eindeutigen Lösungen von zwei gekoppelten gewöhnlichen DGLen. Die Kauf- und Verkaufsdichten sind jeweils als eindeutige Lösungen von linearen hyperbolischen PDGLen, die anhand der Erwartungswerte der einkommenden Orderparameter festgelegt sind, gegeben. Die Lösungen sind in geschlossener Form erhältlich. Im zweiten Teil wird ein funktionaler zentraler Grenzwertsatz d.h. ein Invarianzprinzip für ein vereinfachtes Modell eines Limitorderbuches bewiesen. Unter einer natürlichen Skalierung konvergiert der zweidimensionale Preisprozess (Bid- und Askpreis) in Verteilung zu einer Semimartingal reflektierten Brownschen Bewegung in der zugelassenen Preismenge. Gleichzeitig konvergieren die Kauf- und Verkaufsdichten im schwachen Sinn zum Betrag einer zweiparametrischen Brownschen Bewegung. Es wird weiterhin anhand eines Beispiels gezeigt, wie man für das Modell im ersten Teil eine stochastiche PDGL, unter einer starken Stationaritätsannahme für die Orderplatzierungen und -stornierungen, herleiten kann. Im dritten Teil wird ein Mittelungs- bzw. ein Invarianzprinzip für diskrete Banach- bzw. Hilbertraumwertige stochastische Prozesse bewiesen.In the first part of the thesis, we define a random state-dependent discrete model of a two-sided limit order book in terms of its key quantities best bid [ask] price and the standing buy [sell] volume density. For a simple scaling that introduces a slow time scaling, that is equivalent to the classical law of large numbers, for the bid/ask prices and a faster time scale for the limit volume placements/cancelations, that keeps the expected volume rate over the considered price interval invariant, we prove a limit theorem. The limit theorem states that, given regularity conditions on the random order flow, the key quantities converge in the sense of a strong law of large numbers to a tractable continuous limiting model. The limiting model is such that the best bid and ask price dynamics can be described in terms of two coupled ODE:s, while the dynamics of the relative buy and sell volume density functions are given as the unique solutions of two linear first-order hyperbolic PDE:s with variable coefficients, specified by the expectation of the order flow parameters. In the second part, we prove a functional central limit theorem i.e. an invariance principle for an order book model with block shaped volume densities close to the spread. The weak limit of the two-dimensional price process (best bid and ask price) is given by a semi-martingale reflecting Brownian motion in the set of admissible prices. Simultaneously, the relative buy and sell volume densities close to the spread converge weakly to the modulus of a two-parameter Brownian motion. We also demonstrate an example how to easily derive an SPDE for the relative volume densities in a simple case, when a strong stationarity assumption is made on the limit order placements and cancelations for the model suggested in the first part. In the third and final part of the thesis, we prove an averaging and an invariance principle for discrete processes taking values in Banach and Hilbert spaces, respectively.
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Bender, Martin. "Limit theorems for generalizations of GUE random matrices." Doctoral thesis, KTH, Matematik (Inst.), 2008. http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-4799.
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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
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Reed, Matthew. "The Central Limit Theorem for Linear Spectral Statistics of Submatrices of the Gaussian Wigner Random Matrices." Thesis, University of California, Davis, 2014. http://pqdtopen.proquest.com/#viewpdf?dispub=3646379.
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The main topic addressed here is the joint distribution of spectra of submatrices M(1) and M(2) of large Gaussian Wigner matrices M. A multidimensional central limit theorem for linear statistics of the eigenvalues of submatrices will be proved with explicit formulas for the covariance that relate the spectra to a random surface model known as the Gaussian free field. The regularity assumption is that test functions belong to the Sobolev space H s, for s > 5/2.
The organization is as follows. Chapters 1 and 2 consist of an introduction to Wigner matrices and the central limit theorem in the random matrix theory. Chapter 3 is a discussion of the results which motivated this work, in addition to an introduction to the Gaussian free field. Chapter 4 contains the new results of the author, and chapter 5 is an appendix describing some technical tools.
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Ysusi, Mendoza Carla Mariana. "Estimation of the variation of prices using high-frequency financial data." Thesis, University of Oxford, 2005. http://ora.ox.ac.uk/objects/uuid:1b520271-2a63-428d-b5a0-e7e9c4afdc66.
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When high-frequency data is available, realised variance and realised absolute variation can be calculated from intra-day prices. In the context of a stochastic volatility model, realised variance and realised absolute variation can estimate the integrated variance and the integrated spot volatility respectively. A central limit theory enables us to do filtering and smoothing using model-based and model-free approaches in order to improve the precision of these estimators. When the log-price process involves a finite activity jump process, realised variance estimates the quadratic variation of both continuous and jump components. Other consistent estimators of integrated variance can be constructed on the basis of realised multipower variation, i.e., realised bipower, tripower and quadpower variation. These objects are robust to jumps in the log-price process. Therefore, given adequate asymptotic assumptions, the difference between realised multipower variation and realised variance can provide a tool to test for jumps in the process. Realised variance becomes biased in the presence of market microstructure effect, meanwhile realised bipower, tripower and quadpower variation are more robust in such a situation. Nevertheless there is always a trade-off between bias and variance; bias is due to market microstructure noise when sampling at high frequencies and variance is due to the asymptotic assumptions when sampling at low frequencies. By subsampling and averaging realised multipower variation this effect can be reduced, thereby allowing for calculations with higher frequencies.
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Lima, Amanda de. "Cohomologia e propriedades estocásticas de transformações expansoras e observáveis lipschitzianos." Universidade de São Paulo, 2007. http://www.teses.usp.br/teses/disponiveis/55/55135/tde-08052007-135433/.
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Provamos o Teorema do Limite Central para transformações expansoras por pedaços em um intervalo e observáveis com variação limitada. Utilizamos a abordagem desenvolvida por R. Rousseau-Egele, como apresentada por A. Broise. O método da demonstração se baseia no estudo de pertubações do operador de transferência de Ruelle-Perron-Frobenius. Uma contribuição original é dada no último capítulo, onde provamos que, para transformações markovianas expansoras, todos os observáveis não constantes, contínuos e com variação limitada não são infinitamente cohomólogos à zero, generalizando um resultado de Bamón, Rivera-Letelier, Urzúa and Kiwi para observáveis lipschitzianos e transformações \'z POT. n\' . A demonstração se baseia na teoria dos operadores de Ruelle-Perron-Frobenius desenvolvida nos capítulos anterioresWe prove the Central Limit Theorem for piecewise expanding interval transformations and observables with bounded variation, using the approach of J.Rousseau-Egele as described by A. Broise. This approach makes use of pertubations of the so-called Ruelle-Perron-Frobenius transfer operator. An original contribution is given in the last chapter, where we prove that for Markovian expanding interval maps all observables which are non constant, continuous and have bounded variation are not infinitely cohomologous with zero, generalizing a result by Bamón, Rivera-Letelier, Urzúa and Kiwi for Lipschitzian observables and the transformations \'z POT. n\' . Our demosntration uses the theory of Ruelle-Perron-Frobenius operators developed in the previos chapters
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Koukkous, Abdellatif. "Comportement hydrodynamique de différents processus de zéro range." Rouen, 1997. http://www.theses.fr/1997ROUES051.
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Cette thèse comprend trois parties. Dans la première, nous étudions le comportement asymptotique d'un processus de zéro range symétrique en volume fini dans un environnement aléatoire. Nous démontrons que, pour presque tout environnement, la mesure empirique converge en probabilité vers l'unique solution faible d'une équation de diffusion non linéaire indépendante de l'environnement. Dans la seconde partie, réalisée en collaboration avec G. Gielis et C. Landim, nous abordons le problème des fluctuations à l'équilibre pour le processus de zéro range avec environnement aléatoire. Il s'agit d'obtenir un résultat de type théorème central limite pour le champ de densité, autrement dit de montrer que le champ des fluctuations de la densité converge en loi vers un champ gaussien généralisé. Nous établissons le principe de Boltzmann-Gibbs et la convergence faible du champ de fluctuations de la densité du processus de zéro range en environnement aléatoire vers un processus d'Ornstein-Uhlenbeck généralisé dont l'évolution est décrite par linéarisation de l'équation hydrodynamique autour d'une densité fixée en présence d'un bruit blanc. Dans la dernière partie, réalisée en collaboration avec O. Benois et C. Landim, nous donnons une nouvelle interprétation des corrections de Navier-Stokes à l'équation hydrodynamique d'un système asymétrique de particules en interaction. Nous considérons un système dont la mesure initiale est associée à un profil constant dans la direction de la dérive. Nous montrons que, sous une renormalisation diffusive, le comportement du processus est décrit par une équation parabolique non linéaire dont le coefficient de diffusion coïncide avec le coefficient de diffusion de l'équation hydrodynamique de la version symétrique du processus.
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Min, Aleksey. "Limit theorems for statistical functionals with applications to dimension estimation." Doctoral thesis, [S.l.] : [s.n.], 2004. http://webdoc.sub.gwdg.de/diss/2004/min/min.pdf.
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Poinas, Arnaud. "Statistiques asymptotiques des processus ponctuels déterminantaux stationnaires et non stationnaires." Thesis, Rennes 1, 2019. http://www.theses.fr/2019REN1S024/document.
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Ce manuscrit est dédié à l'étude de l'estimation paramétrique d'une famille de processus ponctuels appelée processus déterminantaux. Ces processus sont utilisés afin de générer et modéliser des configurations de points possédant de la dépendance négative, dans le sens où les points ont tendance à se repousser entre eux. Plus précisément, nous étudions les propriétés asymptotiques de divers estimateurs classiques de processus déterminantaux paramétriques, stationnaires et non-stationnaires, dans les cas où l'on observe une unique réalisation d'un tel processus sur une fenêtre bornée. Ici, l'asymptotique se fait sur la taille de la fenêtre et donc, indirectement, sur le nombre de points observés. Dans une première partie, nous montrons un théorème limite central pour une classe générale de statistiques sur les processus déterminantaux. Dans une seconde partie, nous montrons une inégalité de béta-mélange générale pour les processus ponctuels que nous appliquons ensuite aux processus déterminantaux. Dans une troisième partie, nous appliquons le théorème limite central obtenu à la première partie à une classe générale de fonctions estimantes basées sur des méthodes de moments. Finalement, dans la dernière partie, nous étudions le comportement asymptotique du maximum de vraisemblance des processus déterminantaux. Nous donnons une approximation asymptotique de la log-vraisemblance qui est calculable numériquement et nous étudions la consistance de son maximumThis manuscript is devoted to the study of parametric estimation of a point process family called determinantal point processes. These point processes are used to generate and model point patterns with negative dependency, meaning that the points tend to repel each other. More precisely, we study the asymptotic properties of various classical parametric estimators of determinantal point processes, stationary and non stationary, when considering that we observe a unique realization of such a point process on a bounded window. In this case, the asymptotic is done on the size of the window and therefore, indirectly, on the number of observed points. In the first chapter, we prove a central limit theorem for a wide class of statistics on determinantal point processes. In the second chapter, we show a general beta-mixing inequality for point processes and apply our result to the determinantal case. In the third chapter, we apply the central limit theorem showed in the first chapter to a wide class of moment-based estimating functions. Finally, in the last chapter, we study the asymptotic behaviour of the maximum likelihood estimator of determinantal point processes. We give an asymptotic approximation of the log-likelihood that is computationally tractable and we study the consistency of its maximum
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Aquino, Juan Carlos, and Gabriel Rodríguez. "Understanding the Functional Central Limit Theorems with Some Applications to Unit Root Testing with Structural Change." Economía, 2013. http://repositorio.pucp.edu.pe/index/handle/123456789/117824.
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The application of different unit root statistics is by now a standard practice in empirical work. Even when it is a practical issue, these statistics have complex nonstandard distributions depending on functionals of certain stochastic processes, and their derivations represent a barrier even for many theoretical econometricians. These derivations are based on rigorous and fundamental statistical tools which are not (very) well known by standard econometricians. This paper aims to fill this gap by explaining in a simple way one of these fundamental tools: namely, the Functional Central Limit Theorem. To this end, this paper analyzes the foundations and applicability of two versions of the Functional Central Limit Theorem within the framework of a unit root with a structural break. Initial attention is focused on the probabilistic structure of the time series to be considered. Thereafter, attention is focused on the asymptotic theory for nonstationary time series proposed by Phillips (1987a), which is applied by Perron (1989) to study the effects of an (assumed) exogenous structural break on the power of the augmented Dickey-Fuller test and by Zivot and Andrews (1992) to criticize the exogeneity assumption and propose a method for estimating an endogenous breakpoint. A systematic method for dealing with efficiency issues is introduced by Perron and Rodriguez (2003), which extends the Generalized Least Squares detrending approach due to Elliot et al. (1996). An empirical application is provided.Hoy en día es una práctica estándar de trabajo empírico la aplicación de diferentes estadísticos de contraste de raíz unitaria. A pesar de ser un aspecto práctico, estos estadísticos poseen distribuciones complejas y no estándar que dependen de funcionales de ciertos procesos estocásticos y sus derivaciones representan una barrera incluso para varios econometristas teóricos. Estas derivaciones están basadas en herramientas estadísticas fundamentales y rigurosas que no son (muy) bien conocidas por econometristas estándar. El presente artículo completa esta brecha al explicar en una forma simple una de estas herramientas fundamentales la cual es el Teorema del Límite Central Funcional. Por lo tanto, este documento analiza los fundamentos y la aplicabilidad de dos versiones del Teorema del Límite Central Funcional dentro del marco de una raíz unitaria con un quiebre estructural. La atención inicial se centra en la estructura probabilística de las series de tiempo propuesta por Phillips (1987a), la cual es aplicada por Perron (1989) para estudiar los efectos de un quiebre estructural (asumido) exógeno sobre la potencia de las pruebas Dickey-Fuller aumentadas y por Zivot y Andrews (1992) para criticar el supuesto de exogeneidad y proponer un método para estimar un punto de quiebre endógeno. Un método sistemático para tratar con aspectos de eficiencia es introducido por Perron y Rodríguez (2003), el cual extiende el enfoque de Mínimos Cuadrados Generalizados para eliminar los componentes determinísticos de Elliot et al. (1996). Se presenta además una aplicación empírica.
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Hurth, Tobias. "Limit theorems for a one-dimensional system with random switchings." Thesis, Georgia Institute of Technology, 2010. http://hdl.handle.net/1853/37201.
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We consider a simple one-dimensional random dynamical system with two driving vector fields and random switchings between them. We show that this system satisfies a one force - one solution principle and compute its unique invariant density explicitly. We study the limiting behavior of the invariant density as the switching rate approaches zero and infinity and derive analogues of classical probabilistic results such as the central limit theorem and large deviations principle.
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Barrera, David. "Quenched Asymptotics for the Discrete Fourier Transforms of a Stationary Process." University of Cincinnati / OhioLINK, 2016. http://rave.ohiolink.edu/etdc/view?acc_num=ucin1460652609.
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Bui, Thi Thuy. "Limit theorems for branching random walks and products of random matrices." Thesis, Lorient, 2020. https://tel.archives-ouvertes.fr/tel-03261556.
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L’objectif du sujet de ma thèse est d’établir des théorèmes limites pour des marches aléatoires avec branchement gouvernées par des produits de matrices aléatoires, en profitant des progrès récents sur les produits de matrices aléatoires et en y établissant de nouveaux résultats adaptés au besoin. La première partie concerne le modèle classique d'une marche aléatoire avec branchement sur la droite réelle. Nous établissons une borne Berry-Esseen et une asymptotique précise de déviation modérée de type Cramér pour la mesure de comptage qui compte le nombre de particules de n-ième génération situées dans une région donnée. La deuxième partie est consacrée à l'étude des produits $G_n = A_n \ldots A_1$ de matrices aléatoires réelles $A_i$ de type $d \times d$, indépendantes et identiquement distribuées. Dans cette partie, avec une motivation pour des applications aux marches aléatoires avec branchement gouvernées par des produits de matrices aléatoires, nous améliorons et étendons le théorème central limite et le théorème limite local établis par Le Page (1982). Dans la troisième partie, on considère un modèle de marches aléatoires avec branchement, où les mouvements des individus sont gouvernés par des produits de matrices aléatoires de type $d \times d$. A l'aide des résultats établis à la deuxième partie pour les produits de matrices aléatoires, on établit un théorème central limite et une expansion asymptotique à grande déviation de type Bahadur-Rao pour la mesure de comptage $ Z_n^x $ qui compte le nombre de particules de n-ième génération situées dans une région donnée avec normalisation appropriée. La quatrième partie est une suite de la troisième partie. Dans cette partie, on établit la borne de type Berry-Esseen à propos de la vitesse de convergence dans le théorème central limite et une asymptotique précise de déviation modérée de type Cramér pour $ Z_n^x $The main objective of my thesis is to establish limit theorems for a branching random walk with products of random matrices by taking advantage of recent advances in products of random matrices and establishing new results as needed. The first part concerns the classic branching random walk on the real line. We establish a Berry- Esseen bound and a Cramér type moderate deviation expansion for the counting measure which counts the number of particles of nth generation situated in a given region. The second part is devoted to the study of the products $G_n = A_n \ldots A_1$ of real random matrices $A_i$ of type $ d \times d$, independent and identically distributed. In this part, with a motivation for applications to branching random walks governed by products of random matrices, we improve and extend the central limit theorem and the local limit theorem established by Le Page (1982). In the third part, we consider a branching random walk model, where the movements of individuals are governed by products of random matrices of type $ d \times d $. Using the results established in the second part for the products of random matrices, we establish a central limit theorem and a large deviation asymptotic expansion of the Bahadur-Rao type for the counting measure $ Z_n^x $ which counts the number n-th generation particles located in a given region with suitable norming. The fourth part is a continuation of the third part. In this part, we establish the Berry-Esseen bound which gives the speed of convergence in the central limit theorem and a precise Cramér- type moderate deviation asymptotic for $ Z_n^x $
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Ahn, Jae Youn. "Non-parametric inference of risk measures." Diss., University of Iowa, 2012. https://ir.uiowa.edu/etd/2808.
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Responding to the changes in the insurance environment of the past decade, insurance regulators globally have been revamping the valuation and capital regulations. This thesis is concerned with the design and analysis of statistical inference procedures that are used to implement these new and upcoming insurance regulations, and their analysis in a more general setting toward lending further insights into their performance in practical situations. The quantitative measure of risk that is used in these new and upcoming regulations is the risk measure known as the Tail Value-at-Risk (T-VaR). In implementing these regulations, insurance companies often have to estimate the T-VaR of product portfolios from the output of a simulation of its cash flows. The distributions for the underlying economic variables are either estimated or prescribed by regulations. In this situation the computational complexity of estimating the T-VaR arises due to the complexity in determining the portfolio cash flows for a given realization of economic variables. A technique that has proved promising in such settings is that of importance sampling. While the asymptotic behavior of the natural non-parametric estimator of T-VaR under importance sampling has been conjectured, the literature has lacked an honest result. The main goal of the first part of the thesis is to give a precise weak convergence result describing the asymptotic behavior of this estimator under importance sampling. Our method also establishes such a result for the natural non-parametric estimator for the Value-at-Risk, another popular risk measure, under weaker assumptions than those used in the literature. We also report on a simulation study conducted to examine the quality of these asymptotic approximations in small samples.
The Haezendonck-Goovaerts class of risk measures corresponds to a premium principle that is a multiplicative analog of the zero utility principle, and is thus of significant academic interest. From a practical point of view our interest in this class of risk measures arose primarily from the fact that the T-VaR is, in a sense, a minimal member of the class. Hence, a study of the natural non-parametric estimator for these risk measures will lend further insights into the statistical inference for the T-VaR. Analysis of the asymptotic behavior of the generalized estimator has proved elusive, largely due to the fact that, unlike the T-VaR, it lacks a closed form expression. Our main goal in the second part of this thesis is to study the asymptotic behavior of this estimator. In order to conduct a simulation study, we needed an efficient algorithm to compute the Haezendonck-Goovaerts risk measure with precise error bounds. The lack of such an algorithm has clearly been noticed in the literature, and has impeded the quality of simulation results. In this part we also design and analyze an algorithm for computing these risk measures. In the process of doing we also derive some fundamental bounds on the solutions to the optimization problem underlying these risk measures. We also have implemented our algorithm on the R software environment, and included its source code in the Appendix.
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Boyer, Jean-Baptiste. "Le théorème central limite pour la marche linéaire sur le tore et le théorème de renouvellement dans Rd." Thesis, Bordeaux, 2016. http://www.theses.fr/2016BORD0075/document.
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La première partie de cette thèse porte sur l’étude de la marche aléatoire sur le tore Td := Rd/Zd définie par une mesure de probabilité SLd(Z). Pour étudier le Théorème Central Limite et la loi du logarithme itéré, nous appliquons la méthode de Gordin qui consiste à se ramener à des martingales. Pour cela, nous utilisons un résultat de Bourgain, Furmann, Lindenstrauss et Mozes nous permettant de résoudre l’équation de Poisson pour des points ayant de bonnes propriétés diophantiennes. Dans la deuxième partie, nous étudions la marche sur Rd\{0} définie par l’action de SLd(R) et nous montrons un résultat de vitesse de convergence dans le théorème de renouvellement de Guivarc’h et Le PageThe first part of this thesis deals with the random walk on the torus Td := Rd/Zd defined by a robability measure on SLd(Z). To study the Central Limit Theorem and the Law of the Iterated Logarithm, we apply Gordin’s method. To do so, we use a result proved by Bourgain, Furmann, Lindenstrauss and Mozes to solve Poisson’s equation at point’s having good diophantine properties.In the second part, we study the walk on Rd \ {0} defined by the action of SLd(R) and we prove a result about the rate of convergence in Guivarc’h and Le Page’s renewal theorem
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Beering, Carina Verfasser], Anne [Akademischer Betreuer] Leucht, Jens-Peter [Akademischer Betreuer] [Kreiß, and Carsten [Akademischer Betreuer] Jentsch. "A Functional Central Limit Theorem and its Bootstrap Analogue for Locally Stationary Processes with Application to Independence Testing / Carina Beering ; Anne Leucht, Jens-Peter Kreiß, Carsten Jentsch." Braunschweig : Technische Universität Braunschweig, 2021. http://d-nb.info/122853375X/34.
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Rodrigues, Chang Kuo. "O teorema central do limite: um estudo ecológico do saber e do didático." Pontifícia Universidade Católica de São Paulo, 2009. https://tede2.pucsp.br/handle/handle/11426.
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Previous issue date: 2009-12-02This 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
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Pesaresi, Emanuele. "Leptokurtic signals in random control vibration testing." Master's thesis, Alma Mater Studiorum - Università di Bologna, 2017.
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In several industrial sectors, some components are subjected to mechanical vibrations which may lead to a premature failure. To ensure that they operate properly during their service life, the utilization of qualification tests has been consolidated over the years. It is often required to carry out accelerated tests for obvious reasons as feasibility and cost: the aim is to limit the duration of tests. The Test Tailoring procedure requires an appropriate definition for vibratory test profiles to be utilized as an excitation in terms of motion generated by vibrating tables or shakers. The synthesis of such profiles requires that signals be measured in real environments and then that their most important characteristics be reproduced in a laboratory, in particular their spectral content and damage potential.The conventional procedures permit the synthesis of an accelerated test profile in terms of a Power Spectral Density, which is characterized by a Gaussian distribution of the corresponding timeseries values. Such a kind of synthesis might be unfit to represent the real environment signal taken as a reference, owing to the latter’s usual non-Gaussianity. As a consequence, reliability could be compromised since the “nature” of the real signal is not preserved. Typical examples of non-Gaussian signals coming forth in real applications are the so-called Leptokurtic signals, whose high amplitude peaks originate a strongly non Gaussian probability distribution. A parameter called kurtosis is often employed to represent the number and severity of the peaks of the signal. A common reference is made to “kurtosis control” whenever it is required that the synthesized and the measured signal have not only the same spectral content but the same kurtosis value as well. In this work some novel Mission Synthesis algorithms are proposed, which generate test profiles by controlling precisely the kurtosis value and complying with the spectral content of the reference signal.
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Passos, Frederico Salgueiro. "O teorema das seções de Lévy aplicado à séries temporais correlacionadas não estacionárias: uma análise da convergência gaussiana em sistemas dinâmicos." Universidade Federal de Alagoas, 2014. http://www.repositorio.ufal.br/handle/riufal/1725.
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Weakly nonstationary processes appear in many challenging problems related to the physics of complex systems. An interesting question is how to quantify the rate of convergence to Gaussian behavior of rescaled heteroscedastic comming from economics time series with stationary first moments but nonstationary multifractal long-range correlated second moments and also time series generated from fractionated brownian motion where the series correlation is dependent of a parameter. Here it is used the approach Which uses a recently proposed extension of the Lévy sections theorem. It was analyzed the statistical and multifractal properties of heteroscedastic time series and found that the Lévy sections approach provides a faster convergence to Gaussian behavior relative to the convergence of traditional partial sums of variables. To understand this transition it is used several statistical tests to provide enough data on convergence behavior. It was also observed that the rescaled signals retain multifractal properties even after reaching what appears to be the stable Gaussian regime.Coordenação de Aperfeiçoamento de Pessoal de Nível Superior
Processos não-estacionários com interações fracas aparecem como problemas desafiadores em sistemas complexos em física. Uma questão interessante é como quantificar a taxa de convergência para o comportamento gaussiano em séries temporais heteroscedásticas, sem uma variância única em toda a série, provenientes de sistemas financeiros, reescaladas com os primeiros momentos estacionários mas com uma multifractalidade não estacionária e segundos momentos que possuem uma correlação do longo alcance e verificar o mesmo mecanismo também em séries temporais geradas a partir de um movimento Browniano Fracionado onde a correlação da série depende de um parâmetro ajustável. Aqui é usada uma extensão do teorema das seções de Lévy. Analisando as propriedades estatísticas e multifractais de uma série temporal heteroscedástica e encontrando que as seções de Lévy fornece uma convergência mais rápida para o comportamento gaussiano relativo à convergência das tradicionais somas de variáveis, o teorema do limite central. Para entender essa transição foram utilizados vários testes estatísticos que forneceram dados suficientes sobre o comportamento de convergência. Também observou-se que os sinais reescalados mantêm suas propriedades multifractais mesmo depois de atingirem um regime que parece ser um regime gaussiano.
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Corker, Lloyd A. "A test for Non-Gaussian distributions on the Johannesburg stock exchange and its implications on forecasting models based on historical growth rates." University of Western Cape, 2002. http://hdl.handle.net/11394/7447.
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Masters of CommerceIf share price fluctuations follow a simple random walk then it implies that forecasting models based on historical growth rates have little ability to forecast acceptable share price movements over a certain period. The simple random walk description of share price dynamics is obtained when a large number of investors have equal probability to buy or sell based on their own opinion. This simple random walk description of the stock market is in essence the Efficient Market Hypothesis, EMT. EMT is the central concept around which financial modelling is based which includes the Black-Scholes model and other important theoretical underpinnings of capital market theory like mean-variance portfolio selection, arbitrage pricing theory (APT), security market line and capital asset pricing model (CAPM). These theories, which postulates that risk can be reduced to zero sets the foundation for option pricing and is a key component in financial software packages used for pricing and forecasting in the financial industry. The model used by Black and Scholes and other models mentioned above are Gaussian, i.e. they exhibit a random nature. This Gaussian property and the existence of expected returns and continuous time paths (also Gaussian properties) allow the use of stochastic calculus to solve complex Black- Scholes models. However, if the markets are not Gaussian then the idea that risk can be. (educed to zero can lead to a misleading and potentially disastrous sense of security on the financial markets. This study project test the null hypothesis - share prices on the JSE follow a random walk - by means of graphical techniques such as symmetry plots and Quantile-Quantile plots to analyse the test distributions. In both graphical techniques evidence for the rejection of normality was found. Evidenceleading to the rejection of the hypothesis was also found through nonparametric or distribution free methods at a 1% level of significance for Anderson-Darling and Runs test.
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Bureaux, Julien. "Méthodes probabilistes pour l'étude asymptotique des partitions entières et de la géométrie convexe discrète." Thesis, Paris 10, 2015. http://www.theses.fr/2015PA100160/document.
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Cette thèse se compose de plusieurs travaux portant sur l'énumération et le comportement asymptotique de structures combinatoires apparentées aux partitions d'entiers. Un premier travail s'intéresse aux partitions d'entiers bipartites, qui constituent une généralisation bidimensionnelle des partitions d'entiers. Des équivalents du nombre de partitions sont obtenus dans le régime critique où l'un des entiers est de l'ordre du carré de l'autre entier et au delà de ce régime critique. Ceci complète les résultats établis dans les années cinquante par Auluck, Nanda et Wright. Le deuxième travail traite des chaînes polygonales à sommets entiers dans le plan. Pour un modèle statistique introduit par Sinaï, une représentation intégrale exacte de la fonction de partition est donnée. Ceci conduit à un équivalent du nombre de chaînes joignant deux points distants qui fait intervenir les zéros non triviaux de la fonction zêta de Riemann. Une analyse combinatoire détaillée des chaînes convexes est présentée. Elle permet de montrer l'existence d'une forme limite pour les chaînes convexes aléatoires ayant peu de sommets, répondant ainsi à une question ouverte de Vershik. Un troisième travail porte sur les zonotopes à sommets entiers en dimension supérieure. Un équivalent simple est donné pour le logarithme du nombre de zonotopes contenus dans un cône convexe et dont les extrémités sont fixées. Une loi des grands nombres est établie et la forme limite est caractérisée par la transformée de Laplace du côneThis thesis consists of several works dealing with the enumeration and the asymptotic behaviour of combinatorial structures related to integer partitions. A first work concerns partitions of large bipartite integers, which are a bidimensional generalization of integer partitions. Asymptotic formulæ are obtained in the critical regime where one of the numbers is of the order of magnitude of the square of the other number, and beyond this critical regime. This completes the results established in the fifties by Auluck, Nanda, and Wright. The second work deals with lattice convex chains in the plane. In a statistical model introduced by Sinaï, an exact integral representation of the partition function is given. This leads to an asymptotic formula for the number of chains joining two distant points, which involves the non trivial zeros of the Riemann zeta function. A detailed combinatorial analysis of convex chains is presented. It makes it possible to prove the existence of a limit shape for random convex chains with few vertices, answering an open question of Vershik. A third work focuses on lattice zonotopes in higher dimensions. An asymptotic equality is given for the logarithm of the number of zonotopes contained in a convex cone and such that the endings of the zonotope are fixed. A law of large numbers is established and the limit shape is characterized by the Laplace transform of the cone
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Ranciaro, Neto Adhemar. "Estudo de séries de tempo financeiras sob a perspectiva do teorema das seções de Lévy." Universidade Federal de Alagoas, 2013. http://www.repositorio.ufal.br/handle/riufal/1656.
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This study aimed to analyze financial time series grounded on a perspective of time measure
changing, based on accumulation of volatility of returns relative to the prices observed. Such
a scale was used for two reasons: the first one is related to Ludwig Von Mises’ proposition of
time concept in an economic system and the second one is related to the acceleration of
convergence in Gaussian distribution of a sequence of random variables, according to Lévy
sections theorem. By means of implementation of this new timeline, we designed a type of
trading asset strategy which its resulting average returns and risk were compared to a strategy
using daily time unit. Results suggested reflection about statistical and measurement
procedures applied to the data.O objetivo deste trabalho foi o de estudar séries temporais financeiras fundamentadas em uma perspectiva de alteração de medida de tempo, baseada no acúmulo de volatilidade dos retornos relativos aos preços observados. Esta escala foi utilizada por dois motivos: o primeiro está relacionado à proposta de Ludwig von Mises sobre a ideia de tempo em um sistema econômico e o segundo está associado à capacidade que tal medida tem de acelerar o processo de convergência de distribuição de uma sequência de variáveis aleatórias para a Gaussiana, de acordo com o teorema das seções de Lévy. Com base nesta nova escala temporal, foi elaborado um tipo de estratégia de negociação de ativos tendo seus retornos médios e risco sido avaliados em comparação com uma estratégia utilizando o tempo em unidades diárias. Os resultados obtidos motivaram a reflexão sobre as estatísticas utilizadas e os procedimentos para a mensuração de desempenho de cada estratégia.
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Filipciuc, Cristina. "Análise da evolução das empresas por separação de observadores." Master's thesis, Instituto Superior de Economia e Gestão, 2020. http://hdl.handle.net/10400.5/21083.
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Mestrado em Métodos Quantitativos para a Decisão Económica e EmpresarialO uso de dados de séries temporais na modelação de redes financeiras e económicas desafiam algumas suposições estatísticas tradicionais, como a aplicação do Teorema de Limite Central (TLC). No entanto, o recurso aos pressupostos provenientes da Física foi possível resolver algumas das limitações abordadas ao longo do documento. O trabalho desenvolvido baseia-se na aplicação de algoritmos de separação de observadores desenvolvidos pela Closer, cujo problema tem sido abordado desde há alguns anos, permitindo assim resolver os problemas associados à aplicação do TLC. Estes algoritmos baseiam-se na geometria diferencial e relatividade, que foram aplicados em séries de ações das empresas do mercado americano retiradas em escalas de tempo diversas, reportando no final os resultados obtidos em termos de transformação das distribuições vistas por cada um dos observadores.
The handling of time-series data in modeling financial and economic networks challenges some traditional statistical assumptions, such as application of the Central Limit Theorem. However, using the assumptions of Physics it was possible to understand some of the limitations of the model. The work developed is set up on the application of observer separation algorithms developed by Closer Consulting, whose problem has addressed for some years, which solves the infinite variation limitation. These algorithms based on differential geometry and relativity, which was applied to series of shares of companies in the American market taken at different time scales, reporting, in the end, the results obtained in terms of transforming the distributions seen by each of the observers.
info:eu-repo/semantics/publishedVersion
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Kreacic, Eleonora. "Some problems related to the Karp-Sipser algorithm on random graphs." Thesis, University of Oxford, 2017. http://ora.ox.ac.uk/objects/uuid:3b2eb52a-98f5-4af8-9614-e4909b8b9ffa.
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We study certain questions related to the performance of the Karp-Sipser algorithm on the sparse Erdös-Rényi random graph. The Karp-Sipser algorithm, introduced by Karp and Sipser [34] is a greedy algorithm which aims to obtain a near-maximum matching on a given graph. The algorithm evolves through a sequence of steps. In each step, it picks an edge according to a certain rule, adds it to the matching and removes it from the remaining graph. The algorithm stops when the remining graph is empty. In [34], the performance of the Karp-Sipser algorithm on the Erdös-Rényi random graphs G(n,M = [cn/2]) and G(n, p = c/n), c > 0 is studied. It is proved there that the algorithm behaves near-optimally, in the sense that the difference between the size of a matching obtained by the algorithm and a maximum matching is at most o(n), with high probability as n → ∞. The main result of [34] is a law of large numbers for the size of a maximum matching in G(n,M = cn/2) and G(n, p = c/n), c > 0. Aronson, Frieze and Pittel [2] further refine these results. In particular, they prove that for c < e, the Karp-Sipser algorithm obtains a maximum matching, with high probability as n → ∞; for c > e, the difference between the size of a matching obtained by the algorithm and the size of a maximum matching of G(n,M = cn/2) is of order Θlog n(n1/5), with high probability as n → ∞. They further conjecture a central limit theorem for the size of a maximum matching of G(n,M = cn/2) and G(n, p = c/n) for all c > 0. As noted in [2], the central limit theorem for c < 1 is a consequence of the result of Pittel [45]. In this thesis, we prove a central limit theorem for the size of a maximum matching of both G(n,M = cn/2) and G(n, p = c/n) for c > e. (We do not analyse the case 1 ≤ c ≤ e). Our approach is based on the further analysis of the Karp-Sipser algorithm. We use the results from [2] and refine them. For c > e, the difference between the size of a matching obtained by the algorithm and the size of a maximum matching is of order Θlog n(n1/5), with high probability as n → ∞, and the study [2] suggests that this difference is accumulated at the very end of the process. The question how the Karp-Sipser algorithm evolves in its final stages for c > e, motivated us to consider the following problem in this thesis. We study a model for the destruction of a random network by fire. Let us assume that we have a multigraph with minimum degree at least 2 with real-valued edge-lengths. We first choose a uniform random point from along the length and set it alight. The edges burn at speed 1. If the fire reaches a node of degree 2, it is passed on to the neighbouring edge. On the other hand, a node of degree at least 3 passes the fire either to all its neighbours or none, each with probability 1/2. If the fire extinguishes before the graph is burnt, we again pick a uniform point and set it alight. We study this model in the setting of a random multigraph with N nodes of degree 3 and α(N) nodes of degree 4, where α(N)/N → 0 as N → ∞. We assume the edges to have i.i.d. standard exponential lengths. We are interested in the asymptotic behaviour of the number of fires we must set alight in order to burn the whole graph, and the number of points which are burnt from two different directions. Depending on whether α(N) » √N or not, we prove that after the suitable rescaling these quantities converge jointly in distribution to either a pair of constants or to (complicated) functionals of Brownian motion. Our analysis supports the conjecture that the difference between the size of a matching obtained by the Karp-Sipser algorithm and the size of a maximum matching of the Erdös-Rényi random graph G(n,M = cn/2) for c > e, rescaled by n1/5, converges in distribution.
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Flenghi, Roberta. "Théorème de la limite centrale pour des fonctionnelles non linéaires de la mesure empirique et pour le rééchantillonnage stratifié." Electronic Thesis or Diss., Marne-la-vallée, ENPC, 2023. http://www.theses.fr/2023ENPC0051.
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Cette thèse porte sur le théorème de la limite centrale, l'un des deux théorèmes limites fondamentaux de la théorie des probabilités avec la loi forte des grands nombres. Le théorème de la limite centrale usuel qui porte sur des fonctionnelles linéaires de la mesure empirique de vecteurs aléatoires indépendants et identiquement distribués a récemment été étendu à des fonctionnelles non linéaires par l'utilisation de la dérivée fonctionnelle linéaire sur l'espace de Wasserstein des mesures de probabilité. Nous généralisons cette extension à la mesure empirique de vecteurs aléatoires indépendants mais non identiquement distribués d'une part et à la mesure empirique des états successifs d'une chaîne de Markov ergodique d'autre part. Dans un second temps, nous nous intéressons au rééchantillonnage stratifié qui est couramment utilisé dans les filtres particulaires. Nous prouvons un théorème de la limite centrale pour le premier rééchantillonnage sous l'hypothèse que les positions initiales des particules sont indépendantes et identiquement distribuées et leurs poids sont proportionnels à une fonction positive des positions qui envoie leur loi commune sur une probabilité possédant une composante non nulle absolument continue par rapport à la mesure de Lebesgue. Ce résultat repose sur la convergence en loi de la partie fractionnaire des sommes partielles de poids normalisés vers une variable aléatoire uniforme sur [0,1]. Plus généralement, nous montrons la convergence en loi vers un vecteur aléatoire uniforme sur [dollar][0,1]^q[dollar] de q sommes partielles d'une suite de variables aléatoires i.i.d. de carré intégrable multipliées par une fonction de la moyenne empirique de cette suite. Pour traiter le couplage introduit par ce facteur commun, nous supposons que la loi commune a une composante non nulle absolument continue par rapport à la mesure de Lebesgue, ce qui assure la convergence en variation totale dans le théorème de la limite centrale pour cette suite. Sous l'hypothèse que la convergence en loi de la partie fractionnaire des poids normalisés reste vraie au étapes suivantes d'un filtre particulaire calculé en alternant des étapes de rééchantillonnage suivant le mécanisme stratifié et des mutations suivant des noyaux markoviens, nous obtenons une formule de récurrence pour la variance asymptotique des particules après n étapes. Nous vérifions la validité de cette formule au travers d'expériences numériquesThis thesis is dedicated to the central limit theorem which is one of the two fundamental limit theorems in probability theory with the strong law of large numbers.The central limit theorem which is well known for linear functionals of the empirical measure of independent and identically distributed random vectors, has recently been extended to non-linear functionals. The main tool permitting this extension is the linear functional derivative, one of the notions of derivation on the Wasserstein space of probability measures.We generalize this extension by first relaxing the equal distribution assumptionand then the independence property to be able to deal with the successive values of an ergodic Markov chain.In the second place, we focus on the stratified resampling mechanism.This is one of the resampling schemes commonly used in particle filters. We prove a central limit theorem for the first resampling according to this mechanism under the assumption that the initial positions are independent and identically distributed and the weights proportional to a positive function of the positions such that the image of their common distribution by this function has a non zero component absolutely continuous with respect to the Lebesgue measure. This result relies on the convergence in distribution of the fractional part of partial sums of the normalized weights to some random variable uniformly distributed on [0,1]. More generally, we prove the joint convergence in distribution of q variables modulo one obtained as partial sums of a sequence of i.i.d. square integrable random variables multiplied by a common factor given by some function of an empirical mean of the same sequence. The limit is uniformly distributed over [dollar][0,1]^q[dollar]. To deal with the coupling introduced by the common factor, we assume that the common distribution of the random variables has a non zero component absolutely continuous with respect to the Lebesgue measure, so that the convergence in the central limit theorem for this sequence holds in total variation distance.Under the conjecture that the convergence in distribution of fractional parts to some uniform random variable remains valid at the next steps of a particle filter which alternates selections according to the stratified resampling mechanism and mutations according to Markov kernels, we provide an inductive formula for the asymptotic variance of the resampled population after n steps. We perform numerical experiments which support the validity of this formula
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Pillala, Lavanya. "Use Of Web-Based Lessons Of Statistical Concepts With Graphics And Animation To Enhance The Effectiveness Of Learning." Wright State University / OhioLINK, 2010. http://rave.ohiolink.edu/etdc/view?acc_num=wright1268779325.
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Abdelkader, Mohamed. "Théorèmes limites dans l'analyse statistique des systèmes dynamiques." Thesis, Toulon, 2017. http://www.theses.fr/2017TOUL0010/document.
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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 bornéeIn 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
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Liu, Chenguang. "Statistical inference for a partially observed interacting system of Hawkes processes." Thesis, Sorbonne université, 2019. http://www.theses.fr/2019SORUS203.
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Nous observons les actions d'un sous-échantillon de K de N d’individus, pendant un intervalle de temps de longueur t>0, pour certaines grandes K≤N. Nous modélisons les relations des individus par i.i.d. Bernoulli (p) variables aléatoires, où p∈(0,1] est un paramètre inconnu. Le taux d’action de chaque individu dépend d’un paramètre inconnu μ>0 et sur la somme de quelque fonction ϕ des âges des actions des individus qui l'influencent. La fonction ϕ est inconnue mais nous supposons qu'elle se désintègre rapidement. Le but de cette thèse est d'estimer le paramètre p, qui est la principale caractéristique du graphe d’interaction, dans l'asymptotique où taille de la population N→∞, la taille de la population observée K→∞, et dans un temps long t→∞. Soit mt le nombre moyen d'actions par individu jusqu'au temps t, qui dépend de tous les paramètres du modèle. Dans le cas sous-critique, où mt augmente linéairement, nous construisons un estimateur de p avec le taux de convergence 1K√+NmtK√+NKmt√. Dans le cas supercritique, où mt augmente rapidement de façon exponentielle, nous construisons un estimateur de p avec le taux de convergence 1K√+NmtK√. Dans un second temps, nous étudions la normalité asymptotique de ces estimateurs. Dans le cas sous-critique, le travail est très technique mais assez général, et nous sommes amenés à étudier trois régimes possibles, en fonction du terme dominant dans 1K√+NmtK√+NKmt√ à 0. Dans le cas supercritique, nous supposons malheureusement quelques conditions supplémentaires et considérons seulement l'un des deux régimes possiblesWe observe the actions of a K sub-sample of N individuals, during some time interval with length t>0, for some large K≤N. We model the relationships of individuals by i.i.d. Bernoulli(p) random variables, where p∈(0,1] is an unknown parameter. The rate of action of each individual depends on some unknown parameter μ>0 and on the sum of some function ϕ of the ages of the actions of the individuals which influence him. The function ϕ is unknown but we assume it rapidly decays. The aim of this thesis is to estimate the parameter p, which is the main characteristic of the interaction graph, in the asymptotic where the population size N→∞, the observed population size K→∞, and in large time t→∞. Let mt be the average number of actions per individual up to time t, which depends on all the parameters of the model. In the subcritical case, where mt increases linearly, we build an estimator of p with the rate of convergence \frac{1}{\sqrt{K}}+\frac{N} m_t\sqrt{K}}+\frac{N}{K\sqrt{m_t}}. In the supercritical case, where mt increases exponentially fast, we build an estimator of p with the rate of convergence 1K√+NmtK√. In a second time, we study the asymptotic normality of those estimators. In the subcritical case, the work is very technical but rather general, and we are led to study three possible regimes, depending on the dominating term in 1K√+NmtK√+NKmt√→0. In the supercritical case, we, unfortunately, suppose some additional conditions and consider only one of the two possible regimes
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Kasparavičiūtė, Aurelija. "Paklaidos įvertis Centrinėje ribinėje teoremoje." Master's thesis, Lithuanian Academic Libraries Network (LABT), 2008. http://vddb.library.lt/obj/LT-eLABa-0001:E.02~2008~D_20080619_124043-00846.
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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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Gomez, Garcia José Gregorio. "Théorèmes limites pour des fonctionnelles de clusters d'extrêmes et applications." Thesis, Cergy-Pontoise, 2017. http://www.theses.fr/2017CERG0916/document.
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Cette thèse traite principalement des théorèmes limites pour les processus empiriques de fonctionnelles de clusters d'extrêmes de séquences et champs aléatoires faiblement dépendants. Des théorèmes limites pour les processus empiriques de fonctionnelles de clusters d'extrême de séries temporelles stationnaires sont donnés par Drees & Rootzén [2010] sous des conditions de régularité absolue (ou "ß-mélange"). Cependant, ces conditions de dépendance de type mélange sont très restrictives : elles sont particulièrement adaptées aux modèles dans la finance et dans l'histoire, et elles sont de plus compliquées à vérifier. Généralement, pour d'autres modèles fréquemment rencontré dans les domaines applicatifs, les conditions de mélange ne sont pas satisfaites. En revanche, les conditions de dépendance faible, selon Doukhan and Louhichi [1999] et Dedecker & Prieur [2004a], sont des conditions qui généralisent les notions de mélange et d'association. Elles sont plus simple à vérifier et peuvent être satisfaites pour de nombreux modèles. Plus précisément, sous des conditions faibles, tous les processus causals ou non causals sont faiblement dépendants: les processus Gaussien, associés, linéaires, ARCH(∞), bilinéaires et notamment Volterra entrent dans cette liste. À partir de ces conditions favorables, nous étendons certains des théorèmes limites de Drees & Rootzén [2010] à processus faiblement dépendants. En outre, comme application des théorèmes précédents, nous montrons la convergence en loi de l'estimateur de l'extremogramme de Davis & Mikosch [2009] et l'estimateur fonctionnel de l'indice extrémal de Drees [2011] sous dépendance faible. Nous démontrons un théorème de la valeur extrême pour les champs aléatoires stationnaires faiblement dépendants et nous proposons, sous les mêmes conditions, un critère du domaine d'attraction d'une loi d'extrêmes. Le document se conclue sur des théorèmes limites pour les processus empiriques de fonctionnelles de clusters d’extrêmes de champs aléatoires stationnaires faiblement dépendants, et met en évidence la convergence en loi de l'estimateur d'un extremogramme de processus spatio-temporels stationnaires faiblement dépendants en tant qu'applicationThis thesis deals mainly with limit theorems for empirical processes of extreme cluster functionals of weakly dependent random fields and sequences. Limit theorems for empirical processes of extreme cluster functionals of stationnary time series are given by Drees & Rootzén [2010] under absolute regularity (or "ß-mixing") conditions. However, these dependence conditions of mixing type are very restrictive: on the one hand, they are best suited for models in finance and history, and on the other hand, they are difficult to verify. Generally, for other models common in applications, the mixing conditions are not satisfied. In contrast, weak dependence conditions, as defined by Doukhan & Louhichi [1999] and Dedecker & Prieur [2004a], are dependence conditions which generalises the notions of mixing and association. These are easier to verify and applicable to a wide list of models. More precisely, under weak conditions, all the causal or non-causal processes are weakly dependent: Gaussian, associated, linear, ARCH(∞), bilinear and Volterra processes are some included in this list. Under these conveniences, we expand some of the limit theorems of Drees & Rootzén [2010] to weakly dependent processes. These latter results are used in order to show the convergence in distribution of the extremogram estimator of Davis & Mikosch [2009] and the functional estimator of the extremal index introduced by Drees [2011] under weak dependence. We prove an extreme value theorem for weakly dependent stationary random fields and we propose, under the same conditions, a domain of attraction criteria of a law of extremes. The document ends with limit theorems for the empirical process of extreme cluster functionals of stationary weakly dependent random fields, deriving also the convergence in distribution of the estimator of an extremogram for stationary weakly dependent space-time processes
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Ådahl, Markus. "Random iteration of isometries." Doctoral thesis, Umeå University, Mathematics and Mathematical Statistics, 2004. http://urn.kb.se/resolve?urn=urn:nbn:se:umu:diva-263.
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This thesis consists of four papers, all concerning random iteration of isometries. The papers are:
I. Ambroladze A, Ådahl M, Random iteration of isometries in unbounded metric spaces. Nonlinearity 16 (2003) 1107-1117.
II. Ådahl M, Random iteration of isometries controlled by a Markov chain. Manuscript.
III. Ådahl M, Melbourne I, Nicol M, Random iteration of Euclidean isometries. Nonlinearity 16 (2003) 977-987.
IV. Johansson A, Ådahl M, Recurrence of a perturbed random walk and an iterated function system depending on a parameter. Manuscript.
In the first paper we consider an iterated function system consisting of isometries on an unbounded metric space. Under suitable conditions it is proved that the random orbit {Zn} ∞n=0, of the iterations corresponding to an initial point Z0, “escapes to infinity" in the sense that P(Zn Є K) → 0, as n → ∞ for every bounded set K. As an application we prove the corresponding result in the Euclidean and hyperbolic spaces under the condition that the isometries do not have a common fixed point.
In the second paper we let a Markov chain control the random orbit of an iterated function system of isometries on an unbounded metric space. We prove under necessary conditions that the random orbit \escapes to infinity" and we also give a simple geometric description of these conditions in the Euclidean and hyperbolic spaces. The results generalises the results of Paper I.
In the third paper we consider the statistical behaviour of the reversed random orbit corresponding to an iterated function system consisting of a finite number of Euclidean isometries of Rn. We give a new proof of the central limit theorem and weak invariance principles, and we obtain the law of the iterated logarithm. Our results generalise immediately to Markov chains. Our proofs are based on dynamical systems theory rather than a purely probabilistic approach.
In the fourth paper we obtain a suficient condition for the recurrence of a perturbed (one-sided) random walk on the real line. We apply this result to the study of an iterated function system depending on a parameter and defined on the open unit disk in the complex plane.
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Stewart, Kathryn Lockwood. "On Truncations of Haar Distributed Random Matrices." Case Western Reserve University School of Graduate Studies / OhioLINK, 2019. http://rave.ohiolink.edu/etdc/view?acc_num=case1554279921382029.
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Zeileis, Achim. "A unified approach to structural change tests based on F statistics, OLS residuals, and ML scores." Institut für Statistik und Mathematik, WU Vienna University of Economics and Business, 2005. http://epub.wu.ac.at/714/1/document.pdf.
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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
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Gonchigdanzan, Khurelbaatar. "ALMOST SURE CENTRAL LIMIT THEOREMS." University of Cincinnati / OhioLINK, 2001. http://rave.ohiolink.edu/etdc/view?acc_num=ucin990028192.
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Larsson-Cohn, Lars. "Gaussian structures and orthogonal polynomials." Doctoral thesis, Uppsala : Matematiska institutionen, Univ. [distributör], 2002. http://publications.uu.se/theses/91-506-1535-1/.
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Reding, Lucas. "Contributions au théorème central limite et à l'estimation non paramétrique pour les champs de variables aléatoires dépendantes." Thesis, Normandie, 2020. http://www.theses.fr/2020NORMR049.
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La thèse suivante traite du Théorème Central Limite pour des champs de variables aléatoires dépendantes et de son application à l’estimation non-paramétrique. Dans une première partie, nous établissons des théorèmes centraux limite quenched pour des champs satisfaisant une condition projective à la Hannan (1973). Les versions fonctionnelles de ces théorèmes sont également considérées. Dans une seconde partie, nous établissons la normalité asymptotique d’estimateurs à noyau de la densité et de la régression pour des champs fortement mélangeants au sens de Rosenblatt (1956) ou bien des champs faiblement dépendants au sens de Wu (2005). Dans un premier temps, nous établissons les résultats pour l’estimateur à noyau de la régression introduit par Elizbar Nadaraya (1964) et Geoffrey Watson (1964). Puis, dans un second temps, nous étendons ces résultats à une large classe d’estimateurs récursifs introduite par Peter Hall et Prakash Patil (1994)This thesis deals with the central limit theorem for dependent random fields and its applications to nonparametric statistics. In the first part, we establish some quenched central limit theorems for random fields satisfying a projective condition à la Hannan (1973). Functional versions of these theorems are also considered. In the second part, we prove the asymptotic normality of kernel density and regression estimators for strongly mixing random fields in the sense of Rosenblatt (1956) and for weakly dependent random fields in the sense of Wu (2005). First, we establish the result for the kernel regression estimator introduced by Elizbar Nadaraya (1964) and Geoffrey Watson (1964). Then, we extend these results to a large class of recursive estimators defined by Peter Hall and Prakash Patil (1994)
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Jonsson, Fredrik. "Almost Sure Central Limit Theory." Thesis, Uppsala University, Department of Mathematics, 2007. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-121066.
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Lam, Hoang Chuong. "Les théorèmes limites pour des processus stationnaires." Phd thesis, Université François Rabelais - Tours, 2012. http://tel.archives-ouvertes.fr/tel-00712572.
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Nous étudions la mesure spectrale des transformations stationnaires, puis nous l'utilisons pour étudier le théorème ergodique et le théorème limite central. Nous étudions également les martingales avec une nouvelle preuve du théorème central limite, sans analyse de Fourier. Pour le théorème limite central pour marches aléatoires dans un environnement aléatoire sur la dimension 1, on donne deux méthodes pour l'obtenir: approximation pour une martingale et méthode des moments. La méthode des martingales fait résoudre l'equation de Dirichlet (I −P )h = 0, alors que celle des moments résoudre l'equation de Poisson (I − P )h = f . Enfin, nous pouvons utiliser la deuxième méthode pour prouver la relation d'Einstein pour des diffusions réversibles dans un environnement aléatoire dans une dimension.
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Paditz, Ludwig. "On the error-bound in the nonuniform version of Esseen's inequality in the Lp-metric." Saechsische Landesbibliothek- Staats- und Universitaetsbibliothek Dresden, 2013. http://nbn-resolving.de/urn:nbn:de:bsz:14-qucosa-112888.
Full textAbstract:
The aim of this paper is to investigate the known nonuniform version of Esseen's inequality in the Lp-metric, to get a numerical bound for the appearing constant L. For a long time the results given by several authors constate the impossibility of a nonuniform estimation in the most interesting case δ=1, because the effect L=L(δ)=O(1/(1-δ)), δ->1-0, was observed, where 2+δ, 0<δ<1, is the order of the assumed moments of the considered independent random variables X_k, k=1,2,...,n. Again making use of the method of conjugated distributions, we improve the well-known technique to show in the most interesting case δ=1 the finiteness of the absolute constant L and to prove L=L(1)=<127,74*7,31^(1/p), p>1. In the case 0<δ<1 we only give the analytical structure of L but omit numerical calculations. Finally an example on normal approximation of sums of l_2-valued random elements demonstrates the application of the nonuniform mean central limit bounds obtained hereDas Anliegen dieses Artikels besteht in der Untersuchung einer bekannten Variante der Esseen'schen Ungleichung in Form einer ungleichmäßigen Fehlerabschätzung in der Lp-Metrik mit dem Ziel, eine numerische Abschätzung für die auftretende absolute Konstante L zu erhalten. Längere Zeit erweckten die Ergebnisse, die von verschiedenen Autoren angegeben wurden, den Eindruck, dass die ungleichmäßige Fehlerabschätzung im interessantesten Fall δ=1 nicht möglich wäre, weil auf Grund der geführten Beweisschritte der Einfluss von δ auf L in der Form L=L(δ)=O(1/(1-δ)), δ->1-0, beobachtet wurde, wobei 2+δ, 0<δ<1, die Ordnung der vorausgesetzten Momente der betrachteten unabhängigen Zufallsgrößen X_k, k=1,2,...,n, angibt. Erneut wird die Methode der konjugierten Verteilungen angewendet und die gut bekannte Beweistechnik verbessert, um im interessantesten Fall δ=1 die Endlichkeit der absoluten Konstanten L nachzuweisen und um zu zeigen, dass L=L(1)=<127,74*7,31^(1/p), p>1, gilt. Im Fall 0<δ<1 wird nur die analytische Struktur von L herausgearbeitet, jedoch ohne numerische Berechnungen. Schließlich wird mit einem Beispiel zur Normalapproximation von Summen l_2-wertigen Zufallselementen die Anwendung der gewichteten Fehlerabschätzung im globalen zentralen Grenzwertsatz demonstriert