[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.

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’équation de Dirichlet (I - P)h = 0, alors que celle des moments résoudre l’équation 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. / We study the spectral measure for stationary transformations, and then apply to Ergodic theorem and Central limit theorem. We study also martingale process with a new proof of the central limit theorem without Fourier analysis. For the central limit theorem for random walks in random environment, we give two methods to obtain it: martingale approximation and moments. The method of martingales solves Dirichlet’s equation (I - P)h = 0, and the method of moments solves Poisson’s equation (I - P)h = f. Finally, we can use the second method to prove the Einstein relation for reversible diffusions in random environment in one dimension.