Sinulis, Arthur: Higher order concentration of measure and applications. 2019
Inhalt
- Contents
- 1 Introduction
- 1.1 Concentration of measure
- 1.2 Our contribution
- 1.3 Discussion of related literature
- 1.3.1 Concentration inequalities for general functions
- 1.3.2 Concentration inequalities for polynomials
- 1.3.3 Further topics
- 1.4 Outline of the thesis
- 2 Preliminaries
- 2.1 Notations
- 2.2 Difference operators and functional inequalities
- 2.3 From functional to concentration inequalities
- 2.4 Independent random variables
- 2.5 (Weakly dependent) Spin systems
- 2.6 Examples of weakly dependent spin systems
- 3 Bernstein-type inequalities
- 3.1 General results
- 3.2 Applications
- 3.2.1 Derivations
- 3.2.2 Weakly dependent product measures
- 3.2.3 Symmetric group
- 3.2.4 Homogeneous polynomials in 0,1-random variables
- 3.3 Proofs and auxiliary results
- 4 Concentration inequalities for bounded functions
- 4.1 General results
- 4.2 Applications
- 4.2.1 Deviation inequalities for empirical processes
- 4.2.2 Concentration properties of U-statistics
- 4.2.3 Polynomials in the Ising model
- 4.2.4 Number of triangles in exponential random graph models
- 4.3 Proofs
- 5 Concentration inequalities for polynomials in independent random variables
- 5.1 General results
- 5.2 Applications
- 5.2.1 Euclidean norm of a vector with independent components
- 5.2.2 Projections and distance to a fixed subspace
- 5.2.3 Spectral bound for a product of a fixed and a random matrix
- 5.2.4 Concentration properties for fixed design linear regression
- 5.2.5 Special cases
- 5.3 The multilinear case: Proof of Theorem 5.4
- 5.4 Hanson–Wright-type inequality: Proof of Proposition 5.5
- 5.5 The polynomial case: Proof of Theorem 5.6
- 5.6 The general sub-exponential case: alpha
- A Approximate tensorization of entropy in finite spaces
- B Properties of Orlicz quasinorms
- C LSIs and difference operators
- Open questions
- Bibliography