Sequences Of Random Matrices Modulated By A Discrete-Time Markov Chain

First and foremost, I would sincerely like to express my gratitude to my advisor, Pro-fessor George Yin, for his continued guidance, encouragement, patience, understanding, and support t

SEQUENCES OF RANDOM MATRICES MODULATED

BY A DISCRETE-TIME MARKOV CHAIN

by

HUY NGUYEN DISSERTATION

Submitted to the Graduate School

of Wayne State University, Detroit, Michigan

in partial fulfillment of the requirements

for the degree of

DOCTOR OF PHILOSOPHY

2022 MAJOR: MATHEMATICS Approved By:

———————————————————–

———————————————————– Advisor

07/01/2022

07/01/2022

07/01/2022

To My Family

First and foremost, I would sincerely like to express my gratitude to my advisor, Pro-fessor George Yin, for his continued guidance, encouragement, patience, understanding, and support that he offered me during the past five years Without his generous help and infinite patience, I am afraid that I would never complete the dissertation Personally speaking, he is a great man who is always humble Professionally, he works hard, and gives me much to aspire Being his student has been a great privilege in my life

I would like to thank Professor Pei-Yong Wang Thank you for serving as my PhD co-advisor after Professor George Yin moved to the University of Connecticut in the Fall of

2020 His help encouraged me me to stay at Wayne State University to complete the Ph.D program Additionally, he taught me several courses that enabled me to complete my research I truly enjoy his teaching

I would like to thank Professor Kazuhiko Shinki for serving as a member on my disser-tation committee He taught me several statistics courses which enlarged views about the applications of mathematics to other fields

I would like to thank Professor Tao Huang and Professor Le Yi Wang for serving as mem-bers on my dissertation committee I thank him for his valuable time and kind instruction

Xiaoli Kong, Prof Shereen Schultz, and Prof Christopher Leirstein.

I owe my deepest gratitude to Wayne State University, especially the Department of Mathematics, where I have studied and worked for five years During the graduate study,

I received much help and support from the administrative staff Specifically, I would like

to thank Ms Barbara Malicke, Dr Tiana Bosley, Ms Maria Vujic, Ms Joanne Lewan,

Ms Carla Sylvester, and Ms Terri Renaud for their kind help Also, I would like to show

my appreciation to Prof Hengguang Li, Chair of Mathematics Department; Prof Daniel Isaksen, Director of Graduate program, and many other people who are dedicated to the development and strength of the Mathematics Department

I would like to express my sincere appreciation to my friends in Vietnam, as well as

in America, Mr Yatin Patel, Dr Son Nguyen, Prof Thanh Le, Prof Quang Nguyen, Mr Randall White and his wife Mrs Faith White, as well as Colin and Becky Mansker I am especially thankful for their very kind support and encouragement during these past years Finally, I would like to thank my family for the selfless support My parents and parents-in-law, my wife and my children have always understood and supported me during the challenges and successes of this PhD process I am thankful for that Thank you, and I love you all

TABLE OF CONTENTS

Dedication ii

Acknowledgements iii

Chapter 1 Introduction 1

1.1 Recent Progress 1

1.2 Markov Modulated Sequences 3

1.3 Outline 4

Chapter 2 Stochastic Differential Equations and Markov Chains 6

2.1 Stochastic process 6

2.2 Itô Integrals 8

2.3 Itô Formula and the Martingale Representation Theorem 10

2.3.1 The Martingale Representation Theorem 12

2.4 Stochastic Differential Equations 13

2.5 Discrete-Time Markov Chains 14

2.5.1 Asymptotic Expansions 19

Chapter 3 Problem Formulation 25

3.1 Problem Formulation and Conditions 25

Chapter 6 Further Remarks and Ramification 49

6.1 A Remark on Non-Zero Drift 49

6.2 Ramification 50

6.3 Additional Remarks 55

Appendix A 57

References 60

Abstract 64

Autobiographical Statement 66

ABSTRACT

SEQUENCES OF RANDOM MATRICES MODULATED

BY A DISCRETE-TIME MARKOV CHAIN

by

HUY NGUYEN August 2022

Advisor: Dr George Yin

Co-Advisor: Dr Pei-Yong Wang

Major: Mathematics

Degree: Doctor of Philosophy

In this dissertation, we consider a number of matrix-valued random sequences that are modulated by a discrete-time Markov chain having a finite space Assuming that the state space of the Markov chain is large, our main effort in this work is devoted to reducing the complexity To achieve this goal, our formulation uses time-scale separation of the Markov chain The state-space of the Markov chain is split into subspaces Next, the states of the Markov chain in each subspace are aggregated into a “super” state Then we normalize the matrix-valued sequences that are modulated by the two-time-scale Markov chain Under

formulation