LONG TIME ASYMPTOTICS OF SOME WEAKLY INTERACTING PARTICLE SYSTEMS AND HIGHER ORDER ASYMPTOTICS OF GENERALIZED FIDUCIAL DISTRIBUTION Abhishek Pal Majumder A dissertation submitted to the faculty at the University of North Carolina at Chapel Hill in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the Department of Statistics and Operations Research. Chapel Hill 2015 Approved by: Amarjit S. Budhiraja Jan H. Hannig Shankar Bhamidi Vladas Pipiras Chuanshu Ji c 2015 Abhishek Pal Majumder ALL RIGHTS RESERVED ii ABSTRACT ABHISHEK PAL MAJUMDER: LONG TIME ASYMPTOTICS OF SOME WEAKLY INTERACTING PARTICLE SYSTEMS AND HIGHER ORDER ASYMPTOTICS OF GENERALIZED FIDUCIAL DISTRIBUTION (Under the direction of Amarjit S. Budhiraja and Jan H. Hannig) In probability and statistics limit theorems are some of the fundamental tools that rig- orously justify a proposed approximation procedure. However, typically such results fail to explain how good is the approximation. In order to answer such a question in a precise quan- titative way one needs to develop the notion of convergence rates in terms of either higher order asymptotics or non-asymptotic bounds. In this dissertation, two different problems are studied with a focus on quantitative convergence rates. In first part, we consider a weakly interacting particle system in discrete time, approx- imating a nonlinear dynamical system. We deduce a uniform in time concentration bound for the Wasserstein-1 distance of the empirical measure of the particles and the law of the corresponding deterministic nonlinear Markov process that is obtained through taking the particle limit. Many authors have looked at similar formulations but under a restrictive com- pactness assumption on the particle domain. Here we work in a setting where particles take values in a non-compact domain and study several time asymptotics and large particle limit properties of the system. We establish uniform in time propagation of chaos along with a rate of convergence and also uniform in time concentration estimates. We also study another discrete time system that models active chemotaxis of particles which move preferentially iii towards higher chemical concentration and themselves release chemicals into the medium dynamically modify the chemical field. Long time behavior of this system is studied. Second part of the dissertation is focused on higher order asymptotics of Generalized Fiducial inference. It is a relevant inferential procedure in standard parametric inference where no prior information of unknown parameter is available in practice. Traditionally in Bayesian paradigm, people propose posterior distribution based on the non-informative priors but imposition of any prior measure on parameter space is contrary to the “no-information” belief (according to Fisher’s philosophy). Generalized Fiducial inference is one such remedy in this context where the proposed distribution on the parameter space is only based on the data generating equation. In this part of dissertation we established a higher order expansion of the asymptotic coverage of one-sided Fiducial quantile. We also studied further and found out the space of desired transformations in certain examples, under which the transformed data generating equation yields first order matching Fiducial distribution. iv Baba, Ma, Chordadu and all my teachers. v ACKNOWLEDGMENTS First and foremost, I would like to thank my two advisors: Amarjit Budhiraja and Jan Hannig for their encouragements, guidances and inspirations throughout my graduate life. Apart from being great teachers and providing invaluable inputs to my research, they also helped me grow as a person, enabling me to find a career path for myself. I learned a lot about academic life as much as I did about my impediments. I will remain forever grateful to Amarjit for having patience in me in this process and teaching me the value of discipline, hard work. I am personally grateful to Jan since he introduced me to the topic of Generalized Fiducial Inference and without his constant support and invaluable insights this work would not have been possible. Their teachings will always stay with me for the rest of my career. Furthermore, I take this opportunity to express my gratitude to both of them for financially supporting me from their NSF grants. I am extremely thankful to Shankar for teaching us measure theory in the very first semester with so much care and later for the course on concentration bounds which helps me enormously to develop a broader intuition for solving any problem through the concepts of probability theory. I am grateful to him for numerous discussions on various other research topics of Probability. I will not forget his countless treats specially the one while Prof. B V Rao was visiting here. I would also like to thank Professor Vladas Pipiras and Chuansu Ji for agreeing to be in my dissertation committee in spite of their extremely busy schedules and for vi numerous other things that I had learned from them. I thank all my teachers at UNC, Chapel Hill especially Michael Kosorok, Edward Carlstein; for their invaluable advices, teachings which were extremely useful in my later research and for being generous to me with their encouragements. Besides that, I want to thank all other professors and staff in this small won- derful Statistics department, whose doors were always open for me. Many thanks to Alison Kieber for becoming my savior for answering all my “important” questions. It is imperative that I thank all my fellow graduate students and postdocs who were sig- nificant part of my life in last few years. I should specially mention Subhamay, Xuan, Ruyou, Suman and Louis for all interesting conversations. I am thankful to all my batch-mates espe- cially James, Jenny, Michael, Minghui for sharing their valuable thoughts in my research and presentations. Special thanks to all my officemates Chris, Washington, John, Hwo-young for all my good times and enriching my graduate life experiences. As they say, “There is nothing on this earth more to be prized than true friendship.” I feel really blessed to have so many good friends. Special thanks to my roommates Sayan Das- gupta and Ritwik Chaudhuri for always being with me through the highs and lows. Without Ritwik’s help I may not end up here. I learned so many things from Sayan that this place will fall short to list them. In one sentence my life would not be easier without him. I should men- tion Pourab, Sid da, Anji, Pdr, Sujatro, Tommy, Poulami, Suprateek da, Suman, Swarnava, Sayantan for filling my life with your cheerful presence for last five years. Special thanks to Rinku di and Samarpan da for your warmths and making all of us your extended family. We are so lucky to have you in Chapel Hill. I am also thankful to Anirban Basak, Jyotishka da, Shalini di for all great times that we spent together. I will miss all of you in my next venture. vii I am grateful to all my teachers at Indian Statistical Institute, Kolkata (especially Prof. S C Bagchi, Alok Goswami, Gopal K Basak, Saurabh Ghosh, Mahuya Dutta and all oth- ers) for teaching me statistics and probability as well as building the inherent mathematical foundation. I should specially thank Prof. Debapriya Sengupta for introducing the problem binary matrix simulation constrained by row and column margins which I think changed my notion of research. I would also thank Professor Debapriya Sengupta, Sumitra Purkayastha, Sourabh Bhattacharya, Indranil Mukhopadhyay, Arijit Chakrabarty for their supports during my struggles in JRF days. I would have never reached this far without your helps. I could not close this section before I thank my family for its unconditional love, support and give special thanks to my first teacher in all of my mathematical training, my father Paritosh Pal Majumder. viii TABLE OF CONTENTS LIST OF TABLES .................................... xii 1 INTRODUCTION .................................. 1 1.1 Sketch of the bound for P (W1(LN ; µ) > ") :.................5 1.2 Connection to Our Problem and Other Questions:...............8 1.3 Outline of Part 1.................................9 1.4 Outline of Chapter4.............................. 14 2 LONG TIME ASYMPTOTICS OF SOME WIPS . 16 2.1 Introduction................................... 16 2.2 Model Description............................... 22 2.3 Main Results.................................. 24 2.4 Proofs...................................... 31 2.4.1 Moment Bounds............................ 31 2.4.2 Proof of Proposition 2.3.1....................... 34 2.4.3 Proof of Proposition 2.3.2....................... 37 2.4.4 Proof of Theorem 2.3.1......................... 39 2.4.5 Proof of Corollary 4.2.1........................ 42 2.4.6 Proof of Proposition 2.3.3....................... 43 2.4.7 Proof of Theorem 2.3.2......................... 47 2.4.8 Proof of Theorem 2.3.3......................... 54 2.4.9 Proof of Theorem 2.3.4......................... 60 ix N 2.4.10 Uniform Concentration Bounds for fηn g................ 66 3 AN IP MODEL FOR ACTIVE CHEMOTAXIS . 72 3.1 Introduction................................... 72 3.2 Description of Nonlinear System:....................... 78 3.3 Main Results:.................................. 82 3.3.1 Concentration Bounds:......................... 91 3.4 Proofs :..................................... 94 3.4.1 Proof of Theorem1........................... 101 3.4.2 Proof of Theorem 3.3.1......................... 111 3.4.3 Proof of Theorem 3.3.2......................... 119