Dependence structures in Lévy-type Markov processes

In this dissertation, we examine the positive and negative dependence of infinitely divisible distributions and Lévy-type Markov processes. Examples of infinitely divisible distributions include Poissonian distributions like compound Poisson and α-stable distributions. Examples of Lévy-type Markov processes include Lévy processes and Feller processes, which include a class of jump-diffusions, certain stochastic differential equations with Lévy noise, and subordinated Markov processes. Other examples of Lévy-type Markov processes are time-inhomogeneous Feller evolution systems (FES), which include additive processes. We will provide a tour of various forms of positive dependence, which include association, positive supermodular association (PSA), positive supermodular dependence (PSD), and positive orthant dependence (POD), and more. We will give a history of the characterization of these notions of positive dependence for infinitely divisible distributions, Lévy processes, and certain Feller diffusions. Additionally, we will present our contribution to the characterization of positive dependence for jump-Feller processes, and include applications. We will also characterize positive dependence for general time-inhomogeneous Feller evolution systems and jump-FESs. Finally, we characterize negative association and other forms of negative dependence for infinitely divisible distributions and Lévy processes.