本书采用了不相关的、来自信息论的研究，角度新颖地提出了一种证明中心极限的新方法，并对此进行了全面描述：书中先是读者呈现了嫡和费雪信息概念的基本导论，随后以一系列与它们行为有关的标准测试作为验证。在作者的独特构思与实证下，信息论与中心极限定理两个看似不相干的领域被巧妙地联结起来，实现了跨学科的科研合作。此外，书里还汇编了一些已发表或尚未发表的论文研究成果，展现了技术如何可以给出一个极限定理的统一观点。

Preface
1.Introduction to Information Theory
1.1 Entropy and relative entropy
1.1.1 Discrete entropy
1.1.2 Differential entropy
1.1.3 Relative entropy
1.1.4 Other entropy-like quantities
1.1.5 Axiomatic definition of entropy
1.2 Link to thermodynamic entropy
1.2.1 Definition of thermodynamic entropy
1.2.2 Maximum entropy and the Second Law
1.3 Fisher information
1.3.1 Definition and properties
1.3.2 Behaviour on convolution
1.4 Previous information-theoretic proofs
1.4.1 Rényi's method
1.4.2 Convergence of Fisher information
2.Convergence in Relative Entropy
2.1 Motivation
2.1.1 Sandwich inequality
2.1.2 Projections and adjoints
2.1.3 Normal case
2.1.4 Results of Brown and Barron
2.2 Generalised bounds on projection eigenvalues
2.2.1 Projection of functions in L
2.2.2 Restricted Poincaré constants
2.2.3 Convergence of restricted Poincaré constants
2.3 Rates of convergence
2.3.1 Proof of O(1/n) rate of convergence
2.3.2 Comparison with other forms of convergence
2.3.3 Extending the Cramér-Rao lower bound
3.Non-Identical Variables and Random Vectors
3.1 Non-identical random variables
3.1.1 Previous results
3.1.2 Improved projection inequalities
3.2 Random vectors
3.2.1 Definitions
3.2.2 Behaviour on convolution
3.2.3 Projection inequalities
4.Dependent Random Variables
4.1 Introduction and notation
4.1.1 Mixing coefficients
4.1.2 Main results
4.2 Fisher information and convolution
4.3 Proof of subadditive relations
4.3.1 Notation and definitions
4.3.2 _ Bounds on densities
4.3.3 Bounds on tails
4.3.4 Control of the mixing coefficients
5.Convergence to Stable Laws
5.1 Introduction to stable laws
5.1.1 Definitions
5.1.2 Domains of attraction
5.1.3 Entropy of stable laws
5.2 Parameter estimation for stable distributions
5.2.1 Minimising relative entropy
5.2.2 Minimising Fisher information distance
5.2.3 Matching logarithm of density
5.3 Extending de Brujjn's identity
5.3.1 Partial differential equations
5.3.2 Derivatives of relative entropy
5.3.3 Integral form of the identities
5.4 Relationship between forms of convergence
5.5 Steps towards a Brown inequality
6.Convergence on Compact Groups
6.1 Probability on compact groups
6.1.1 Introduction to topological groups
6.1.2 Convergence of convolutions
6.1.3 Conditions for uniform convergence
6.2 Convergence in relative entropy
6.2.1 Introduction and results
6.2.2 Entropy on compact groups
6.3 Comparison of forms of convergence
6.4 Proof of convergence in relative entropy
6.4.1 Explicit rate of convergence
6.4.2 No explicit rate of convergence
7.Convergence to the Poisson Distribution
7.1 Entropy and the Poisson distribution
7.1.1 The law of small numbers
7.1.2 Simplest bounds on relative entropy
7.2 Fisher information
7.2.1 Standard Fisher information
7.2.2 Scaled Fisher information
7.2.3 Dependent variables
7.3 Strength of bounds
7.4 De Bruijn identity
7.5 L2 bounds on Poisson distance
7.5.1 L2 definitions
7.5.2 Sums of Bernoulli variables
7.5.3 Normal convergence
8.Free Random Variables
8.1 Introduction to free variables
8.1.1 Operators and algebras
8.1.2 Expectations and Cauchy transforms
8.2 Derivations and conjugate functions
8.2.1 Derivations
8.2.2 Fisher information and entropy
8.3 Projection inequalities
Appendix A Calculating Entropies
A.1 Gamma distribution
A.2 Stable distributions
Appendix B Poincaré Inequalities
B.1 Standard Poincaré inequalities
B.2 Weighted Poincaré inequalities
Appendix C de Bruijn Identity
Appendix D Entropy Power Inequality
Appendix E Relations