This book describes extensions of Sudakov's classical result on the concentration of measure phenomenon for weighted sums of dependent random variables. The central topics of the book are weighted sums of random variables and the concentration of their distributions around Gaussian laws. The analysis takes place within the broader context of concentration of measure for functions on high-dimensional spheres. Starting from the usual concentration of Lipschitz functions around their limiting mean, the authors proceed to derive concentration around limiting affine or polynomial functions, aiming towards a theory of higher order concentration based on functional inequalities of log-Sobolev and Poincaré type. These results make it possible to derive concentration of higher order for weighted sums of classes of dependent variables.
While the first part of the book discusses the basic notions and results from probability and analysis which are needed for the remainder of the book, the latter parts provide a thorough exposition of concentration, analysis on the sphere, higher order normal approximation and classes of weighted sums of dependent random variables with and without symmetries.
Author(s): Sergey Bobkov, Gennadiy Chistyakov, Friedrich Götze
Series: Probability Theory and Stochastic Modelling, 104
Publisher: Springer
Year: 2023
Language: English
Pages: 437
City: Cham
Preface
Contents
Part I Generalities
Chapter 1 Moments and Correlation Conditions
1.1 Isotropy
1.2 First Order Correlation Condition
1.3 Moments and Khinchine-type Inequalities
1.4 Moment Functionals Using Independent Copies
1.5 Variance of the Euclidean Norm
1.6 Small Ball Probabilities
1.7 Second Order Correlation Condition
Chapter 2 Some Classes of Probability Distributions
2.1 Independence
2.2 Pairwise Independence
2.3 Coordinatewise Symmetric Distributions
2.4 Logarithmically Concave Measures
2.5 Khinchine-type Inequalities for Norms and Polynomials
2.6 One-dimensional Log-concave Distributions
2.7 Remarks
Chapter 3 Characteristic Functions
3.1 Smoothing
3.2 Berry–Esseen-type Inequalities
3.3 Lévy Distance and Zolotarev’s Inequality
3.4 Lower Bounds for the Kolmogorov Distance
3.5 Remarks
Chapter 4 Sums of Independent Random Variables
4.1 Cumulants
4.2 Lyapunov Coefficients
4.3 Rosenthal-type Inequalities
4.4 Normal Approximation
4.5 Expansions for the Product of Characteristic Functions
4.6 Higher Order Approximations of Characteristic Functions
4.7 Edgeworth Corrections
4.8 Rates of Approximation
4.9 Remarks
Part II Selected Topics on Concentration
Chapter 5 Standard Analytic Conditions
5.1 Moduli of Gradients in the Continuous Setting
5.2 Perimeter and Co-area Inequality
5.3 Poincaré-type Inequalities
5.4 The Euclidean Setting
5.5 Isoperimetry and Cheeger-type Inequalities
5.6 Rothaus Functionals
5.7 Standard Examples and Conditions
5.8 Canonical Gaussian Measures
5.9 Remarks
Chapter 6 Poincaré-type Inequalities
6.1 Exponential Integrability
6.2 Growth of ??-norms
6.3 Moment Functionals. Small Ball Probabilities
6.4 Weighted Poincaré-type Inequalities
6.5 The Brascamp–Lieb Inequality
6.6 Coordinatewise Symmetric Log-concave Distributions
6.7 Remarks
Chapter 7 Logarithmic Sobolev Inequalities
7.1 The Entropy Functional and Relative Entropy
7.2 Definitions and Examples
7.3 Exponential Bounds
7.4 Bounds Involving Relative Entropy
7.5 Orlicz Norms and Growth of ??-norms
7.6 Bounds Involving Second Order Derivatives
7.7 Remarks
Chapter 8 Supremum and Infimum Convolutions
8.1 Regularity and Analytic Properties
8.2 Generators
8.3 Hamilton–Jacobi Equations
8.4 Supremum/Infimum Convolution Inequalities
8.5 Transport-Entropy Inequalities
8.6 Remarks
Part III Analysis on the Sphere
Chapter 9 Sobolev-type Inequalities
9.1 Spherical Derivatives
9.2 Second Order Modulus of Gradient
9.3 Spherical Laplacian
9.4 Poincaré and Logarithmic Sobolev Inequalities
9.5 Isoperimetric and Cheeger-type Inequalities
9.6 Remarks
Chapter 10 Second Order Spherical Concentration
10.1 Second Order Poincaré-type Inequalities
10.2 Bounds on the ?2-norm in the Euclidean Setup
10.3 First Order Concentration Inequalities
10.4 Second Order Concentration
10.5 Second Order Concentration With Linear Parts
10.6 Deviations for Some Elementary Polynomials
10.7 Polynomials of Fourth Degree
10.8 Large Deviations for Weighted ℓ?-norms
10.9 Remarks
Chapter 11 Linear Functionals on the Sphere
11.1 First Order Normal Approximation
11.2 Second Order Approximation
11.3 Characteristic Function of the First Coordinate
11.4 Upper Bounds on the Characteristic Function
11.5 Polynomial Decay at Infinity
11.6 Remarks
Part IV First Applications to Randomized Sums
Chapter 12 Typical Distributions
12.1 Concentration Problems for Weighted Sums
12.2 The Structure of Typical Distributions
12.3 Normal Approximation for Gaussian Mixtures
12.4 Approximation in Total Variation
12.5 ??-distances to the Normal Law
12.6 Lower Bounds
12.7 Remarks
Chapter 13 Characteristic Functions of Weighted Sums
13.1 Upper Bounds on Characteristic Functions
13.2 Concentration Functions of Weighted Sums
13.3 Deviations of Characteristic Functions
13.4 Deviations in the Symmetric Case
13.5 Deviations in the Non-symmetric Case
13.6 The Linear Part of Characteristic Functions
13.7 Remarks
Chapter 14 Fluctuations of Distributions
14.1 The Kantorovich Transport Distance
14.2 Large Deviations for the Kantorovich Distance
14.3 Pointwise Fluctuations
14.4 The Lévy Distance
14.5 Berry–Esseen-type Bounds
14.6 Preliminary Bounds on the Kolmogorov Distance
14.7 BoundsWith a Standard Rate
14.8 Deviation Bounds for the Kolmogorov Distance
14.9 The Log-concave Case
14.10 Remarks
Part V Refined Bounds and Rates
Chapter 15 ?2 Expansions and Estimates
15.1 General Approximations
15.2 Bounds for ?2-distance With a Standard Rate
15.3 ExpansionWith Error of Order ?−1
15.4 Two-sided Bounds
15.5 Asymptotic Formulas in the General Case
15.6 General Lower Bounds
Chapter 16 Refinements for the Kolmogorov Distance
16.1 Preliminaries
16.2 Large Interval. Final Upper Bound
16.3 Relations Between Kantorovich, ?2 and Kolmogorov distances
16.4 Lower Bounds
16.5 Remarks
Chapter 17 Applications of the Second Order Correlation Condition
17.1 Mean Value of ?(?? ,?) Under the Symmetry Assumption
17.2 Berry–Esseen Bounds Involving ?
17.3 Deviations Under Moment Conditions
17.4 The Case of Non-symmetric Distributions
17.5 The Mean Value of ?(?? ,?) in the Presence of Poincaré Inequalities
17.6 Deviations of of ?(?? ,?) in the Presence of Poincaré Inequalities
17.7 Relation to the Thin Shell Problem
17.8 Remarks
Part VI Distributions and Coefficients of Special Type
Chapter 18 Special Systems and Examples
18.1 Systems with Lipschitz Condition
18.2 Trigonometric Systems
18.3 Chebyshev Polynomials
18.4 Functions of the Form ?? (?, ?) = ? (?? + ?)
18.5 The Walsh System on the Discrete Cube
18.6 Empirical Measures
18.7 Lacunary Systems
18.8 Remarks
Chapter 19 DistributionsWith Symmetries
19.1 Coordinatewise Symmetric Distributions
19.2 Behavior On Average
19.3 Coordinatewise Symmetry and Log-concavity
19.4 Remarks
Chapter 20 Product Measures
20.1 Edgeworth Expansion for Weighted Sums
20.2 Approximation of Characteristic Functions of Weighted Sums
20.3 Integral Bounds on Characteristic Functions
20.4 Approximation in the Kolmogorov Distance
20.5 Normal Approximation Under the 4-th Moment Condition
20.6 Approximation With Rate ?−3/2
20.7 Lower Bounds
20.8 Remarks
Chapter 21 Product Measures
21.1 Bernoulli Coefficients
21.2 Random Sums
21.3 Existence of Infinite Subsequences of Indexes
21.4 Selection of Indexes from Integer Intervals
References
Glossary
Index
