This book is aimed at graduate students and researchers who are interested in the probability limit theory of random matrices and random partitions. It mainly consists of three parts. Part I is a brief review of classical central limit theorems for sums of independent random variables, martingale sequences and Markov chains, etc. These classical theorems are frequently used in the study of random matrices and random partitions where random matrices are well-studied in probability theory. Part II concentrates on the asymptotic distribution theory of Circular Unitary Ensemble and Gaussian Unitary Ensemble, which are prototypes of random matrix theory. It turns out that the classical central limit theorems and methods are applicable in describing asymptotic distributions of eigenvalue statistics like linear functionals of eigenvalues. This is attributed to the nice algebraic structures of models. This part also studies the Circular βEnsembles and Gaussian βEnsembles, which may be viewed as extensions of the Circular Unitary Ensemble and Gaussian Unitary Ensemble. Part III is devoted to the study of random uniform and Plancherel partitions. As is known, there is a surprising similarity between random matrices and random integer partitions from the viewpoint of asymptotic distribution theory, though it is difficult to find any direct link between the two finite models. This book treats only second-order fluctuations for primary random variables from two classes of special random models. It is written in a clear, concise and pedagogical way. It may be read as an introductory text to further study probability theory of general random matrices, random partitions and even random point processes. This book is aimed at graduate students and researchers who are interested in probability limit theory of random matrices and random integer partitions. Key Features: ○ The book treats only two special models of random matrices, that is, Circular andGaussian Unitary Ensembles, and the focus is on second-order fluctuations of primaryeigenvalue statistics. So all theorems and propositions can be stated and proved in aclear and concise language ○ In a companion part, the book also treats two special models of random integerpartitions, namely, random uniform and Plancherel partitions. It exhibits a surprising similarity between random matrices and random partitions from the viewpoint ofasymptotic distribution theory, though there is no direct link between finite models ○ The limit distributions of most statistics of interest are obtained by reducing toclassical central limit theorems for sums of independent random variables, martingalesequences and Markov chains. So the book is easily accessible to readers that are familiar with a standard probability theory textbook

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