Alguns teoremas limites para sequências de variáveis aleatórias

• Ano de defesa: 2014
• Autor(a) principal: Waiandt, Euclésio Rangel
• Tipo de documento: Dissertação
• Tipo de acesso: Acesso aberto
• Idioma: por
• Instituição de defesa: Universidade Federal do Espírito Santo; BR; Mestrado em Matemática; Centro de Ciências Exatas; UFES; Programa de Pós-Graduação em Matemática
• Palavras-chave em Português: Sequences of Random Variables; Characteristic functions; Infinitely divisible and stable distributions; Limits theorems; Sequências de Variáveis aleatórias; Funções características; Distribuições infinitamente divisíveis e estáveis; Teoremas limites; Variáveis aleatórias; Convoluções (Matemática); Distribuição (Teoria da probabilidade); Matemática; 51
• Link de acesso: http://repositorio.ufes.br/handle/10/1704

The Central Limit Theorem and the Law of Large Numbers are among the most important results of probability theory. The first one seeks conditions under which v????-E???? ?? ???????? converges in distribution to the normal distribution with parameters 0 and 1, when ?? tends to infinity, where ???? is the sum of ?? independent random variables. At the same time, the second gives conditions such that ????-E???? ?? converges to zero, or equivalently, that ???? ?? converges to the expectation of the random variables, if they are identically distributed. In both cases, the sequences discussed are of the type ????+???? ???? , where ???? > 0 and ???? are real constants. Characterizing the possible limits of such sequences is one of the goals of this dissertation, as they not only converge to a degenerated random variable or a random variable with normal distribution, as the Law of Large Numbers and the Central Limit Theorem, respectively. Thus, we are naturally led to the study of infinitely divisible and stable distributions and their limits theorems. This becomes the main objective of this dissertation. In order to prove the theorems, the method of Lyapunov is applied as the main strategy, which analyzes the convergence of the sequence of characteristic functions related to the random variables. So we carry out a detailed approach of such functions in this research.