Information theory and the central limit theorem

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 R¨¦nyi'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 Poincar¨¦ constants 2.2.3 Convergence of restricted Poincar¨¦ 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 Cram¨¦r-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