This dissertation is concerned with the notion of copula and its importance in modeling and estimation. We use the theory of copulas to assess dependence properties of stationary Markov chains and convergence to the Brownian motion. In the introductory chapter we overview the theory of copulas and their relationship with dependence coefficients for Markov chains. In Chapter 1, we investigate the rates of convergence to zero of the dependence coefficients of copula-based Markov chains. We first review some theoretical results, then improve them and propose an estimate of the maximal correlation coefficient between consecutive states. We also comment on the relationship between geometric ergodicity and exponential rho-mixing for reversible Markov chains. We show that convex combinations of (absolutely regular) geometrically ergodic stationary reversible Markov chains are (absolutely regular) geometrically ergodic and exponential rho-mixing. Moreover, we show that this result holds if only one of the summands is geometrically ergodic. Most of the results are based on our observation that if the absolutely continuous part of the copula has a density bounded away from 0 on a set of measure 1, then it generates absolutely regular stationary Markov chains. Many other striking results are provided on this topic in subsequent sections of Chapter 1. We also use small sets to investigate beta-mixing rates for the Frechet and Mardia families of copulas. We provide new copula families with functions as parameters. We derive the copula for the general Metropolis-Hastings algorithm and use it to apply our results to this class of processes. In Chapter 2, we provide a background survey on functional central limit theorem for stationary Markov chains with a general state space. We emphasize the relationship between the dependence coefficients studied in Chapter 1 and convergence of normalized partial sums of the chain to a standard normal random variable. We present results showing that in most of the cases we consider, the invariance principle holds. In Chapter 3, we study the functional central limit theorem for stationary Markov chains with self-adjoint operator and general state space. We investigate the case when the variance of the partial sum is not asymptotically linear in n, and establish that conditional convergence in distribution of partial sums implies functional CLT. The main tools are maximal inequalities that are further exploited to derive conditions for tightness and convergence to the Brownian motion. The maximal inequalities are based on a new forward-backward martingale decomposition with a triangular array of differences.
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