S-Space College of Natural Sciences (자연과학대학) Dept. of Statistics (통계학과) Theses (Master's Degree_통계학과)
Central Limit Theorem in Non-commutative Probability Space
- 자연과학대학 통계학과
- Issue Date
- 서울대학교 대학원
- Non-commutative probability space; Free independence; Central limit theorem; Random matrix theory; Combinatorics
- 학위논문 (석사)-- 서울대학교 대학원 : 자연과학대학 통계학과, 2018. 2. 임요한.
- The assumption of commutativity of random variables is very natural in classical probability theory. However, many objects such as random matrices do not satisfy commutativity. Thus, we need to build a non-commutative probabilistic structure to handle these objects. Then, many properties in classical probability theory differs from those in this new theory, titled ``free probability theory'', including independence and convergence in distribution. In this paper, we introduce algebraically a non-commutative probability space and provides the notion of ``free independence'', which is the analogue of independence in classical probability theory. Furthermore, we prove the free version of Central Limit Theorem by using such new concepts and several tools in combinatorics, and compare it to the classical Central Limit Theorem.