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ii I dedicate my thesis to my parents, my wife Shudan Ma, for their consistent support and unconditional love. iii ACKNOWLEDGMENTS Pursuing math PhD degree in Purdue university has been a truly adventurous and life-changing journey for me. It would not have been possible for me to achieve this goal without the support and guidance that I received from many people. First and foremost, I would like to express my gratitude and thanks to my advisor, Professor Fabrice Baudoin for being a tremendous mentor for me. Throughout my PhD life in Purdue university, he provided consistent encouragement, sound advice, excellent teaching; Having spent the past 6 years as his student, I was impressed by his deep insights on the subject and clear explanation of the problems. I am so lucky to have him as my advisor. I also would like to thank Professor Yip, Professor Figueroa-Lopez and Profes-sor Ouyang for serving as my committee members. I sincerely thank Professor Yip and Professor Figueroa-Lopez for teaching great courses on probability theory, which stimulate my great interest in the subject. It is a great honor to work with Professor Cheng Ouyang, who provide inspiring discussions and constructive suggestions. I am indebted to many of my graduate fellows for their help throughout my years in Purdue. I would like to express my special appreciation and thanks to Botong Wang and Kevin Mugo for being wonderful Qual instructor and prepared me well for my algebra qualifier exams. I also would like to thank Jing Wang, Jishnu Jaganathan, Bumsik Kim for the stimulating discussions on my research subject. My thanks also goes to Kevin Rotz, Reginald McGee for being great oce mates and for all their helps. I also wish to thank all the sta ↵ in math department for their great e↵orts to make a wonderful place for us to work and study. My special thanks go to Rececca Lank and Nancy Eberle who have been great graduate coordinators. iv Last but not least, I would like to thank my parents, my family and all my friends who have always been there with consistent encouragement and spiritual support