Cadeias de Markov: um estudo espectral

Abstract

The aim of this work is to present a study on Markov Chains in discrete time and finite state space. The theme will be approached gradually, in such a way that we will start with basic and essential elements for a general understanding. Next, we will talk about classifications of states and chains, as well as propositions related to this topic that will be important for verification of later results such as the Convergence Theorem in Total Variation. In this theorem, we will see how the transition matrix will converge and the necessary conditions that guarantee convergence. To do so, we will approach a type of distribution that we will call stationary and we will establish a metric of distance between two probability distributions. Finally, we will deal with Mixing Time, where we will introduce a concept that defines the time required for the distance between two probability distributions be sufficiently small. All these concepts discussed will be important for building the relationship between Markov Chain and Spectral Theory in which, based on the results constructed with the study of eigenvalues and eigenvectors of a matrix, we will find limits for distances between probability distributions for certain types of chains in function of the eigenvalues of the transition matrix.