The novelty and originality of the work consists in the fact that in order to obtain asymptotic expansions with optimal values of the remainder terms in the zone of large deviations, along with the cumulant method the classical method of characteristic functions has to be used. In addition, when solving the problems stated in the work, other than the well known results in the problems of limit theorems of the probability theory and mathematical statistics, we have to estimate constants. Technically it is frequently rather a complicated task. The results obtained in the work have good opportunities to be applied in probability theory, mathematical statistics, econometric, etc. That is illustrated in the last section of the work in which theorems of large deviations are proved in the summation of weighted random variables with weights as well as discounted limit theorems.

Partial exchangeability is the fundamental building block in the subjective approach to the probability of multi-type sequences which replaces the independence concept of the objective theory. The aim of this thesis is to present some theory for partially exchangeable sequences of random variables based on well-known results for exchangeable sequences. The reader is introduced to the concepts of partially exchangeable events, partially exchangeable sequences of random variables and partially exchangeable o-fields, followed by some properties of partially exchangeable sequences of random variables. Extending de Finetti's representation theorem for exchangeable random variables to hold for multi-type sequences, we obtain the following result to be used throughout the thesis: There exists a o-field, conditional upon which, an infinite partially exchangeable sequence of random variables behaves like an independent sequence of random variables, identically distributed within types. Posing (i) a stronger requirement (spherical symmetry) and (ii) a weaker requirement (the selection property) than partial exchangeability on the infinite multi-type sequence of random variables, we obtain results related to de Finetti's representation theorem for partially exchangeable sequences of random variables. Regarding partially exchangeable sequences as mixtures of independent and identically distributed (within types) sequences, we (i) give three possible expressions for the directed random measures of the partially exchangeable sequence and (ii) look at three possible expressions for the o-field mentioned in de Finetti's representation theorem. By manipulating random measures and using de Finetti's representation theorem, we point out some concrete ways of constructing partially exchangeable sequences. The main result of this thesis follows by extending de Finetti's represen. tation theorem in conjunction with the Chatterji principle to obtain the following result: Given any a.s. limit theorem for multi-type sequences of independent random variables, identically distributed within types, there exists an analogous theorem satisfied by all partially exchangeable sequences and by all sub-subsequences of some subsequence of an arbitrary dependent infinite multi-type sequence of random variables, tightly distributed within types. We finally give some limit theorems for partially exchangeable sequences of random variables, some of which follow from the above mentioned result. / Thesis (Ph.D.)-University of Natal, Durban, 1991.

Nagrinėjama silpnai priklausomų atsitiktinių dydžių statistika. Šio magistrinio darbo tikslas, turimą sumą, suvesti į nepriklausomų atsitiktinių dydžių sumą, kuri leistų tolimesniam tyrimui, pritaikyti žinomas teoremas, skaičiavimus bei rezultatus iš nepriklausomų atsitiktinių dydžių teorijos. / We analyze of weakly dependent random variables statistics. The objective of this master thesis is to deduce sum to independent random variables sum, which will be useful for applaying known theorems, calculations and results from the theory in independent random variables.

Magistro baigiamajame darbe analizuojami matematinės statistikos tikimybiniai skirstiniai. Darbo tikslas – atlikti nepriklausomų beta atsitiktinių dydžių sandaugos pasiskirstymo analizę. Ši tematika aktuali sudėtingų sistemų patikimumo analizės teorijoje. Darbe aprašyta neaprėžtai dalių tikimybinių skirstinių teorija, kuri naudojama atsitiktinių dydžių sumos pasiskirstymui tirti, ir M-dalių tikimybių pasiskirstymo modelių teorija, naudojama nepriklausomų atsitiktinių dydžių sandaugos pasiskirstymo analizei atlikti. Baigiamajame darbe suformuluotos dvi teoremos, nusakančios, kuriais atvejais daugindami baigtinį skaičių nepriklausomų beta atsitiktinių dydžių su skirtingomis parametrų reikšmėmis gauname beta atsitiktinį dydį, kurio parametrai išreikšti per dauginamųjų parametrus. Šios teoremos įrodytos, kuomet dauginame du nepriklausomus beta atsitiktinius dydžius. Taip pat darbe suformuluota ir įrodyta teorema, nurodanti beta atsitiktinio dydžio logaritmo charakteringosios funkcijos analitinę išraišką. / The mathematical statistics probability distributions are analyzed in this master thesis. The main purpose is to carry out the analysis of independent beta random variables products distribution. This theme is relevant to the reliability analysis of complex systems theory. The theoretics of unlimited divisible probability distributions is described in this master thesis. Following theory is useful to investigate the sums of independent random variables. The theoretics of M-divisible probability distributions is also described in this master thesis. It is useful to investigate the product of independent random variables. Two theorems about the product of finite number of independent beta random variables are formulated in this master thesis. Following theorems tells us, that product is beta random variable again, and its parameters expressions are related with multiplicands parameters. Theorems are proved when we are multiplying two independent beta random variables. Another theorem that is about characteristic function of beta random variable logarithm is formulated and proved in this master thesis.

Thesis (M.S.)--State University of New York at Binghamton, Thomas J. Watson School of Engineering and Applied Science, Department of Computer Science, 2009. / Includes bibliographical references.

[pt] Este trabalho apresenta um método geral e prático, baseado em técnicas de convolução discreta, para avaliar o valor esperado e a variância dos índices de freqüência e duração (FeD) da perda de carga de um sistema de geração. O método proposto modela sem qualquer restrição unidades geradoras com estados operativos de capacidade reduzida simultaneamente com todos possíveis estados de carga. O valor esperado e a variância dos índices de freqüência e duração da perda de carga são calculados, tratando as taxas de transição entre estados de geração e os picos de carga previstos como variáveis aleatórias. Técnicas de agregação são utilizadas para modelar as incertezas das disponibilidades da capacidade de geração das unidades hidráulicas. Sob este aspecto, os índices de confiabilidade são também variáveis aleatórias. A determinação da expansão da capacidade de geração na presença de incertezas é a proposta sob um novo enfoque. Este método avalia limites de confiança ( ou percentis) não só para os índices de perda de carga, mas também para os índices de Freqüência e Duração da falha. Assim, a expansão da capacidade de geração é reutilizada utilizando os percentis dos índices de risco em vez de se utilizar apenas seus valores médios. O método FeD proposto é utilizado para avaliar o impacto das incertezas nos parâmetros dos geradores e da carga, em sistemas típicos da geração, incluindo o Sistema Teste da confiabilidade do IEEE e uma configuração planejada para 1991 do Sistema Sul-Sudeste Brasileiro. / [en] This work presents a practical and general method base on discrete convolution techniques for evaluating the expectation and variance of the frequency and duration (FeD) reliability indices, for a single area system. The proposed method models simultaneously and without any restriction, derated generating capacity generation units with all possible load levels. The expectation and variance of Frequency and duration reability indices are evaluated based on discrete convolution, treating the the generating unit transition rates and forecast peak loads as random variables. Clustering techniques are used to model the incertainties in the output capacities of hydraulic units. Under this interpretation, the reliability indices are also random variables. The generation capacity expansion problem under uncertainties, for a single-area system, is addressed from a new point of view. This method determines confidence limits (or percenties) not only for the loss of load indices but also for the frequency and duration failure indices. The generating capacity expansion is then calculated through risk indices percenties instead of the usual expected values. The proposed FeD method is then used for evaluating the impact of load and generating parameter uncertainties on typical systems including the IEEE Reliability Test System, and the Brazlian South/Southeastern Generating System.

O intuito deste trabalho é fazer uma análise sobre o processo de integração de funções. Existem muitas generalizações do conceito de integração abordado inicialmente por meio da integral de Riemann, como por exemplo, a integral de Riemann-Stieltjes, Lebesgue, Henstock-Kurzweil entre outras. Abordaremos especialmente a integral de Riemann-Stieltjes, e mostraremos a limitação da integral de Riemann no estudo de convergência de funções, indicando a necessidade de se generalizar o processo de integração. Faremos uma aplicação da integral de Riemann-Stieltjes no estudo de variáveis aleatórias e apresentamos uma proposta de abordagem, para a sala de aula, sobre o deslocamento e distância percorrida por um objeto em movimento retilíneo uniforme associado a área. / The aim of this work is analizing the process of integration of functions. There are many generalizations of the integration concept originally addressed by Riemann integral such as the Riemann-Stieltjes integral, Lebesgue integral, Henstock-Kurzweil integral, among others. We will be specially concerned with the integral of Riemann-Stieltjes and we will show the limitations of Riemann integral about convergence of functions, leading to the need to generalize the integration process. We will apply Riemann-Stieltjes integral for the study of random variables and present an approach to the classroom, on the displacement and distance traveled by an object in uniform rectilinear motion associated to concept of area.

This project will focus on finding formulas for E[N(t)] using one of the classical problems in the discipline first, and then extending the scope of the problem to include overall times greater than the time t in the original problem. The expected values in these cases will be found using the uniform and exponential distributions of random variables.

In this thesis, we provide a self contained introduction to the theory of random matrices and matrix models. Our analysis has a chronological order and it begins with the study of nuclear energy levels and ensemble averaging which yields the famous Wigner surmise. Then, the standard Gaussian theory of the orthogonal, unitary and symplectic ensemble is derived from symmetry arguments of the corresponding physical system, which is then followed by the explicit calculation of Wigner's semicircle distribution. Moreover, interactions are introduced to the free theory which leads to the topological expansion, a diagrammatic way of evaluating certain expectation values. Also, it is shown that there exists a duality between the resulting graphs and the quantization of a two dimensional surface through mapping as well as a method for solving a specific family of potentials. Finally, the numerical confirmation for some observables is carried out using the Hamburger moment problem, and the derivation of critical points for some theories.

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