High Dimensional Probability IX: The Ethereal Volume

Preface
Contents
Part I Inequalities and Convexity
Covariance Representations, Lp-Poincaré Inequalities, Stein's Kernels, and High-Dimensional CLTs
1 Introduction
2 Notations and Preliminaries
3 Representation Formulas and Lp-Poincaré Inequalities
4 Stein's Kernels and High-Dimensional CLTs
5 Appendix
References
Volume Properties of High-Dimensional Orlicz Balls
1 Notation and Statement
2 Probabilistic Formulation
3 Probabilistic Preliminaries
4 Proof of Theorem 2.1
5 Application to Spectral Gaps
6 Asymptotic Independence of Coordinates
7 Integrability of Linear Functionals
References
Entropic Isoperimetric Inequalities
1 Introduction
2 Nagy's Theorem
3 One Dimensional Isoperimetric Inequalities for Entropies
4 Special Orders
5 Fisher Information in Higher Dimensions
6 Two Dimensional Isoperimetric Inequalities for Entropies
7 Isoperimetric Inequalities for Entropies in Dimension n=3 and Higher
References
Transport Proofs of Some Functional Inverse Santaló Inequalities
1 Introduction
2 Entropy-Transport and Inverse Santaló Inequalities
2.1 From Entropy-Transport to Inverse Santaló Inequalities
2.2 Different Equivalent Formulations of Inverse Santaló Inequalities
3 Proofs of Entropy-Transport Inequalities in Dimension 1
3.1 The One-Dimensional Symmetric Case
3.2 The One-Dimensional General Case
4 Revisiting the Unconditional Case
Appendix: Proof of Lemma 2.2
References
Tail Bounds for Sums of Independent Two-Sided Exponential Random Variables
1 Introduction
2 Proof of Theorem 1
3 Generalisations
3.1 Examples
3.2 Proof of Theorem 4: The Upper Bound
3.3 Proof of Theorem 4: The Lower Bound
4 Further Remarks
4.1 Moments
4.2 Upper Bounds on Upper Tails from S-Inequalities
4.3 Heavy-Tailed Distributions
4.4 Theorem 1 in a More General Framework
References
Boolean Functions with Small Second-Order Influences on the Discrete Cube
1 Introduction
2 Main Results
3 Auxiliary Notation and Tools
4 Proof of the Main Results
5 Alternative Proof
References
Some Notes on Concentration for α-Subexponential Random Variables
1 Introduction
2 A Generalized Hanson–Wright Inequality
3 Convex Concentration for Random Variables with Bounded Orlicz Norms
4 Uniform Tail Bounds for First- and Second-Order Chaos
5 Random Tensors
Appendix A
References
Part II Limit Theorems
Limit Theorems for Random Sums of Random Summands
1 Introduction and Statement of Results
2 Concentration and Convergence
2.1 General Concentration
2.2 Convergence Conditions
3 Proofs of Main Results
References
A Note on Central Limit Theorems for Trimmed Subordinated Subordinators
1 Introduction
2 Two Methods of Trimming W
3 Self-Standardized CLTs for W
3.1 Self-Standardized CLTs for Method I Trimming
3.2 Self-Standardized CLTs for Method II Trimming
4 Appendix 1
5 Appendix 2
References
Functional Central Limit Theorem via Nonstationary Projective Conditions
1 Introduction and Notations
2 Projective Criteria for Nonstationary Time Series
2.1 Functional CLT Under the Standard Normalization n
2.2 A More General FCLT for Triangular Arrays
3 Applications
3.1 Application to ρ-mixing Triangular Arrays
3.2 Application to Functions of Linear Processes
3.3 Application to the Quenched FCLT
3.4 Application to Locally Stationary Processes
4 The Case of α-Dependent Triangular Arrays
4.1 Application to Functions of α-Dependent Markov Chains
4.2 Application to Linear Statistics with α-Dependent Innovations
4.3 Application to Functions of a Triangular Stationary Markov Chain
References
Part III Stochastic Processes
Sudakov Minoration for Products of Radial-Type Log-ConcaveMeasures
1 Introduction
2 Results
3 Cube-Like Sets
4 How to Compute Moments
5 Positive Process
6 Small Coefficients
7 Large Coefficients
8 The Partition Scheme
References
Lévy Measures of Infinitely Divisible Positive Processes: Examples and Distributional Identities
1 Introduction
2 Preliminaries on Lévy Measures
3 Illustrations
3.1 Poisson Process
3.2 Sato Processes
3.3 Stochastic Convolution
3.4 Tempered Stable Subordinator
3.5 Connection with Infinitely Divisible Random Measures
3.5.1 Cluster Representation
3.5.2 A Characterization of Infinitely Divisible Random Measures
3.5.3 A Decomposition Formula
3.5.4 Some Remarks
3.6 Infinitely Divisible Permanental Processes
4 Transfer of Continuity Properties
5 A Limit Theorem
References
Bounding Suprema of Canonical Processes via Convex Hull
1 Formulation of the Problem
2 Regular Growth of Moments
2.1 γX-Functional
3 Toy Case: 1-Ball
4 Case II. Euclidean Balls
4.1 Counterexample
4.2 4+δ Moment Condition
4.3 Ellipsoids
5 Case III. qn-Balls, 2 6 Concluding Remarks and Open Questions
References
Part IV High-Dimensional Statistics
Random Geometric Graph: Some Recent Developmentsand Perspectives
1 Introduction
1.1 Random Graph Models
1.2 Brief Historical Overview of RGGs
1.3 Outline
2 The Random Geometric Graph Model and Its Variants
2.1 (Soft-) Random Geometric Graphs
2.2 Translation Invariant Random Geometric Graphs
2.3 Markov Random Geometric Graphs
2.4 Other Model Variants
3 Detecting Geometry in RGGs
3.1 Detecting Geometry in the Dense Regime
3.2 Failure to Extend the Proof Techniques to the Sparse Regime
3.3 Toward the Resolution of Geometry Detection
3.3.1 A First Improvement When d>n
3.3.2 Reaching the Polylogarithmic Regime
3.4 Open Problems and Perspectives
4 Nonparametric Inference in RGGs
4.1 Description of the Model and Notations
4.2 Estimating the Matrix of Probabilities
4.3 Spectrum Consistency of the Matrix of Probabilities
4.4 Estimation of the Envelope Function
4.5 Open Problems and Perspectives
5 Growth Model in RGGs
5.1 Description of the Model
5.2 Spectral Convergences
5.3 Estimation Procedure
5.4 Nonparametric Link Prediction
6 Connections with Community-Based Models
6.1 Extension of RGGs to Take into Account Community Structure
6.2 Robustness of Spectral Methods for Community Detection with Geometric Perturbations
6.3 Recovering Latent Positions
6.4 Some Perspectives
Appendix: Outline of the Proofs of Theorems 6 and 7
References
Functional Estimation in Log-Concave Location Families
1 Introduction
2 Main Results
3 Error Bounds for the MLE
4 Concentration Bounds
5 Bias Reduction
6 Minimax Lower Bounds
References