Exchangeable pairs in Stein's method of distributional approximation

UnrestrictedStein's method is a powerful tool used to obtain error terms in distributional approximation and limit theorems. In one formulation, an auxiliary stochastic object termed an exchangeable pair must be created in order to apply the method. This dissertation has two main purposes, the first is to examine the effect that the choice of exchangeable pair has in the computation and quality of the error, and the second is to derive new tools using exchangeable pairs and the main principles of Stein's method.; In examining the method, we will focus on examples of a combinatorial and algebraic nature in theorems pertaining to normal and Poisson approximation. For normal approximation, we analyze the role of the exchangeable pair for the binomial distribution and the Plancherel measure of a random walk on the symmetric group. Poisson approximation for exchangeable pairs is less developed than normal approximation, so that the examples presented are the binomial and negative binomial distributions, both of which have representations as sums of independent random variables.; The new tools developed here include exchangeable pair formulations for Poisson and translated Poisson approximation which are more in the spirit of existing normal approximation theorems than those available. Also, there have been many recent developments adapting Stein's method to approximation by discrete analogs of the normal distribution which contain a smoothing term that is the main technical difficulty in applications. Exchangeable pairs can be used to derive a simple bound on this term that easily reproduces many results found in the literature