Central Limit Theorem in Non-commutative Probability Space

Abstract

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.