University of Tennessee, Knoxville TRACE: Tennessee Research and Creative Exchange Doctoral Dissertations Graduate School 8-2017 Dependence structures in Lévy-type Markov processes Eddie Brendan Tu University of Tennessee, Knoxville, [email protected] Follow this and additional works at: https://trace.tennessee.edu/utk_graddiss Part of the Other Mathematics Commons, and the Probability Commons Recommended Citation Tu, Eddie Brendan, "Dependence structures in Lévy-type Markov processes. " PhD diss., University of Tennessee, 2017. https://trace.tennessee.edu/utk_graddiss/4661 This Dissertation is brought to you for free and open access by the Graduate School at TRACE: Tennessee Research and Creative Exchange. It has been accepted for inclusion in Doctoral Dissertations by an authorized administrator of TRACE: Tennessee Research and Creative Exchange. For more information, please contact [email protected]. To the Graduate Council: I am submitting herewith a dissertation written by Eddie Brendan Tu entitled "Dependence structures in Lévy-type Markov processes." I have examined the final electronic copy of this dissertation for form and content and recommend that it be accepted in partial fulfillment of the requirements for the degree of Doctor of Philosophy, with a major in Mathematics. Jan Rosinski, Major Professor We have read this dissertation and recommend its acceptance: Vasileios Maroulas, Yu-Ting Chen, Haileab Hilafu Accepted for the Council: Dixie L. Thompson Vice Provost and Dean of the Graduate School (Original signatures are on file with official studentecor r ds.) Dependence structures in L´evy-type Markov processes A Dissertation Presented for the Doctor of Philosophy Degree The University of Tennessee, Knoxville Eddie Brendan Tu August 2017 c by Eddie Brendan Tu, 2017 All Rights Reserved. ii I dedicate this dissertation to my parents, Albert and Amy Tu, and to my sister, Dr. Kelly Tu Frantz. My parents are my greatest inspiration. They faced war and communism and rose to the challenges as immigrants in America to give their children an opportunity for a good life. I am so grateful for the sacrifices they've given over the years and for the respect and humility they taught me. My sister Kelly is my greatest role model. She is a wizard in a constant juggling act of extraordinary challenges, which she always seems to overcome. Her passion and dedication to her field of study inspired me to obtain this degree. iii Acknowledgments I would like to begin by thanking my advisor Dr. Jan Rosinski for his patience and guidance over the last fours years. His mentorship changed how I think about mathematics and enabled me to overcome many challenges in my graduate career. Additionally, he is an inspiration as a teacher and presenter, and his willingness and ability to communicate difficult ideas, effectively and intuitively, spurred me to become an educator; for that, I am very grateful. I would also like to thank my committee members Drs. Vasileios Maroulas, Yu-Ting Chen, and Haileab Hilafu. I am very grateful for the willingness of each of you to assist me, answer my questions, and guide me through this process of earning my Ph.D. In particular, I would like to thank Dr. Maroulas for constantly pushing us and challenging us in the classroom so that we could strive to be our best as mathematicians and presenters. A big thank you also goes out to Pam Armentrout for her calming presence and guidance over the past six years. I also want to thank Dr. Marie Jameson for her guidance in the process of helping me get a job and for the many fruitful conversations about teaching. I want to thank the many friends and colleagues whom have been an amazing support system over the past six years: Brian Allen, Peter Jantsch, Tyler Massaro, Nate Pollesch, John Cummings, Kelly Rooker, Steve Fassino, Will Clagett, Marina Massaro, Nick Dexter, Gara Wolf, Joe Daws, Darrin Weber, Andrew Starnes, Andrew Marchese, Chase Worley, Ernest Jum, Kevin Sonnanburg, Greg Clark, and Liguo Wang. I also want to thank my friends back in Virginia: Robbie, Matt, Dustin, Sam, Will, Pamela Palma, and Ray V., for being a constant reminder of keeping a good sense of humor and attitude in the face of difficult challenges. Lastly, and certainly not the least, a huge thanks goes out to my friend Raymond Wodarski. Your courage to overcome extreme difficulties and circumstances, day after day, gives me the strength to take on my challenges. iv \[T]hat I could find company and consolation and hope in an object pulled almost at random from a bookshelf - felt akin to an instance of religious grace." - Jonathan Franzen v Abstract In this dissertation, we examine the positive and negative dependence of infinitely divisible distributions and L´evy-type Markov processes. Examples of infinitely divisible distributions include Poissonian distributions like compound Poisson and α-stable distributions. Examples of L´evy-type Markov processes include L´evyprocesses and Feller processes, which include a class of jump-diffusions, certain stochastic differential equations with L´evynoise, and subordinated Markov processes. Other examples of L´evy-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, L´evyprocesses, 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 L´evyprocesses. vi Table of Contents 1 Introduction1 1.1 Notation......................................8 2 Positive dependence and infinite divisibility 10 2.1 Association, weak association, and positive correlation............ 10 2.2 Positive supermodular association, supermodular dependence, orthant depen- dence........................................ 23 2.2.1 Stochastic orders............................. 23 2.2.2 Dependence induced from stochastic orders............... 25 2.3 Positive dependence and infinite divisibility................... 37 2.3.1 Infinitely divisible (ID) distributions.................. 37 2.3.2 Positive dependence in ID distributions................. 39 2.3.3 Positive dependence and infinite divisibility of squared Gaussians.. 45 3 Positive dependence in L´evy-type Markov processes 46 3.1 Feller processes.................................. 46 3.1.1 L´evyprocesses.............................. 51 3.2 Positive dependence in general Markov processes................ 56 3.3 Positive dependence in L´evyprocesses..................... 61 3.4 Positive dependence in Feller processes..................... 61 3.4.1 Bounded symbols............................. 62 3.4.2 Integro-differential operator, extended generator............ 63 3.4.3 Small-time asymptotics.......................... 65 vii 3.4.4 Main results of this chapter....................... 66 3.5 Applications and examples............................ 86 3.5.1 L´evyprocesses.............................. 86 3.5.2 Ornstein-Uhlenbeck process....................... 86 3.5.3 Feller's pseudo-Poisson process..................... 87 3.5.4 Bochner's subordination of a Feller process............... 90 3.5.5 L´evy-driven stochastic differential equations.............. 92 4 Positive dependence in time-inhomogeneous Markov processes 98 4.1 Time-inhomogeneous Markov processes..................... 98 4.1.1 Time-homogeneous transformation of time-inhomogeneous Markov process................................... 101 4.2 Association of time-inhomogeneous Markov processes............. 106 4.2.1 Temporal association........................... 113 4.2.2 Other forms of dependence in time-inhomogeneous Markov processes 114 4.3 Applications and examples............................ 116 4.3.1 Additive processes............................ 116 4.3.2 Comparison of Markov processes.................... 121 5 Negative dependence in ID distributions and L´evyprocesses 125 5.1 Various forms of negative dependence...................... 125 5.2 Negative dependence and infinite divisibility.................. 133 5.3 Negative dependence in L´evyprocesses..................... 142 5.4 Positive and negative dependence in limit theorems.............. 146 Bibliography 150 Vita 156 viii List of Figures 2.1 Implication map of positive dependence..................... 29 2.2 Gaussian vs. α-stable............................... 45 3.1 Equivalence of dependencies under condition (3.29) for Feller processes... 67 3.2 Transformed L´evymeasure concentration.................... 96 5.1 Implication map of negative dependence.................... 131 5.2 Negative dependence for jump-L´evyprocesses................. 145 ix Chapter 1 Introduction Association is an important property used to study positive dependence in multivariate distributions and stochastic processes. This property has a wide variety of applications, such as being a useful tool to measure system reliability in reliability theory and for proving limit theorems of interacting particle systems in physics. Additionally,