We introduce real second-order freeness in second-order noncommutative probability spaces. We demonstrate that under this definition, independent ensembles of the three real models of random matrices which we consider, namely real Ginibre matrices, Gaussian orthogonal matrices, and real Wishart matrices, are asymptotically second-order free. These ensembles do not satisfy the complex definition of second-order freeness satisfied by their complex analogues. This definition may be used to calculate the asymptotic fluctuations of products of matrices in terms of the fluctuations of each ensemble. We use a combinatorial approach to the matrix calculations similar to genus expansion, but in which nonorientable surfaces appear, demonstrating the commonality between the real ensembles and the distinction from their complex analogues, motivating this distinct definition. We generalize the description of graphs on surfaces in terms of the symmetric group to the nonorientable case. In the real case we find, in addition to the terms appearing in the complex case corresponding to annular spoke diagrams, an extra set of terms corresponding to annular spoke diagrams in which the two circles of the annulus are oppositely oriented, and in which the matrix transpose appears. / Thesis (Ph.D, Mathematics & Statistics) -- Queen's University, 2011-09-09 11:07:37.414

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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.

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;

Thesis (M.S.)--University of Texas at Dallas, 2008. / Includes vita. Includes bibliographical references (leaves 62-63)

In this thesis we study the central limit theorem (CLT) for nonuniformly hyperbolic dynamical systems. We examine cases in which polynomial decay of correlations leads to a CLT with a non-standard scaling factor of √ n ln n. We also formulate an explicit expression for the the diffusion constant σ in situations where a return time function on the system is a certain class of supermartingale. We then demonstrate applications by exhibiting the CLT for the return time function in four classes of dynamical billiards, including one previously unproven case, the skewed stadium, as well as for the linked twist map. Finally, we introduce a new class of billiards which we conjecture are ergodic, and we provide numerical evidence to support that claim.

[pt] 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. / [en] 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.

This thesis is focused around Edgeworths expansion for aproximation of distribution for parameter estimation. Aim of the thesis is to introduce term Edgeworths expansion, its assumptions and terminology associeted with it. Afterwords demonstrate process of deducting first term of Edgeworths expansion. In the end demonstrate this deduction on examples and compare it with different approximations (mainly central limit theorem), and show strong and weak points of Edgeworths expansion.

We consider second and third order limit theorems--namely central-limit theorems, Berry-Esseen bounds and concentration inequalities-- and extend them for "symmetric" random objects, and general estimators of exchangeable structures. At first, we consider random processes whose distribution satisfies a symmetry property. Examples include exchangeability, stationarity, and various others. We show that, under a suitable mixing condition, estimates computed as ergodic averages of such processes satisfy a central limit theorem, a Berry-Esseen bound, and a concentration inequality. These are generalized further to triangular arrays, to a class of generalized U-statistics, and to a form of random censoring. As applications, we obtain new results on exchangeability, and on estimation in random fields and certain network model; extend results on graphon models; give a simpler proof of a recent central limit theorem for marked point processes; and establish asymptotic normality of the empirical entropy of a large class of processes. In certain special cases, we recover well-known properties, which can hence be interpreted as a direct consequence of symmetry. The proofs adapt Stein's method. Subsequently, we consider a sequence of-potentially random-functions evaluated along a sequence of exchangeable structures. We show that, under general stability conditions, those values are asymptotically normal. Those conditions are vaguely reminiscent of those familiar from concentration results, however not identical. We require that the output of the functions does not vary significantly when an entry is disturbed; and the size of this variation should not depend markedly on the other entries. Our result generalizes a number of known results, and as corollaries, we obtain new results for several applications: For randomly sub-sampled subgraphs; for risk estimates obtained by K-fold cross validation; and for the empirical risk of double bagging algorithms. The proof adapts the martingale central-limit theorem.

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