Mathematica is an extremely powerful and flexible symbolic computer algebra system that enables the user to deal with complicated algebraic tasks. It can also easily handle the numerical and graphical sides. One such task is the derivation of moment generating functions (MGF) and characteristic functions (CF), demonstrably effective tools to characterize a distribution. In this paper, we define some rules in Mathematica to help in computing the MGF and CF for linear combination of independent random variables. These commands utilizes pattern-matching code that enhances Mathematica's ability to simplify expressions involving the product of algebraic terms. This enhancement to Mathematica's functionality can be of particular benefit for MGF and CF. Applications of these rules to determine mean, variance and distribution are illustrated for various independent random variables.

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We take a Student process that is based on independent copies of a random variable X and has trajectories in the function space D[0,1]. As a consequence of a functional central limit theorem for this process, with X in the domain of attraction of the normal law, we consider convergence in distribution of several functionals of this process and derive respective asymptotic confidence intervals for the mean of X. We explore the expected lengths and finite-sample coverage probabilities of these confidence intervals and the one obtained from the asymptotic normality of the Student t-statistic, thus concluding some alternatives to the latter confidence interval that are shorter and/or have at least as high coverage probabilities.

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 anteriores / We 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

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.

We consider random simplicial complexes constructed on a Poisson point process within a convex set in a Euclidean space, especially the Vietoris-Rips complex and the Cech complex both of whose 1-skeleton is the Gilbert graph. We investigate at first the Vietoris-Rips complex by considering the volume-power functionals defined by summing powers of the volume of all k-dimensional faces in the complex. The asymptotic behaviour of these functionals is investigated as the intensity of the underlying Poisson point process tends to infinity and the distance parameter goes to zero. This behaviour is observed in different regimes. Univariate and multivariate central limit theorems are proven, and analogous results for the Cech complex are then given. Finally we provide a Poisson limit theorem for the components of the f-vector in the sparse regime.

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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 maximum. / This 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.

<p>This thesis consists of four papers, all concerning random iteration of isometries. The papers are:</p><p>I. Ambroladze A, Ådahl M, Random iteration of isometries in unbounded metric spaces. Nonlinearity 16 (2003) 1107-1117.</p><p>II. Ådahl M, Random iteration of isometries controlled by a Markov chain. Manuscript.</p><p>III. Ådahl M, Melbourne I, Nicol M, Random iteration of Euclidean isometries. Nonlinearity 16 (2003) 977-987.</p><p>IV. Johansson A, Ådahl M, Recurrence of a perturbed random walk and an iterated function system depending on a parameter. Manuscript.</p><p>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 {<i>Z</i>n} ^∞_n=0, of the iterations corresponding to an initial point Z₀, “escapes to infinity" in the sense that <i>P</i>(<i>Z</i>n Є <i>K)</i> → 0, as <i>n</i> → ∞ for every bounded set <i>K</i>. 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.</p><p>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.</p><p>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 <b>R</b>n. 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.</p><p>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. </p>

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 <b>R</b>n. 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.