Bobkov S., Chistyakov G., Götze F. Concentration and Gaussian Approximation for Randomized Sums

Cham: Springer, 2023. — 438 p.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.Preface
Contents
Generalities
Moments and Correlation Conditions
Isotropy
First Order Correlation Condition
Moments and Khinchine-type Inequalities
Moment Functionals Using Independent Copies
Variance of the Euclidean Norm
Small Ball Probabilities
Second Order Correlation Condition
Some Classes of Probability Distributions
Independence
Pairwise Independence
Coordinatewise Symmetric Distributions
Logarithmically Concave Measures
Khinchine-type Inequalities for Norms and Polynomials
One-dimensional Log-concave Distributions
Remarks
Characteristic Functions
Smoothing
Berry–Esseen-type Inequalities
Lévy Distance and Zolotarev’s Inequality
Lower Bounds for the Kolmogorov Distance
Remarks
Sums of Independent Random Variables
Cumulants
Lyapunov Coefficients
Rosenthal-type Inequalities
Normal Approximation
Expansions for the Product of Characteristic Functions
Higher Order Approximations of Characteristic Functions
Edgeworth Corrections
Rates of Approximation
Remarks
Selected Topics on Concentration
Standard Analytic Conditions
Moduli of Gradients in the Continuous Setting
Perimeter and Co-area Inequality
Poincaré-type Inequalities
The Euclidean Setting
Isoperimetry and Cheeger-type Inequalities
Rothaus Functionals
Standard Examples and Conditions
Canonical Gaussian Measures
Remarks
Poincaré-type Inequalities
Exponential Integrability
Growth of 𝑳𝒑-norms
Moment Functionals. Small Ball Probabilities
Weighted Poincaré-type Inequalities
The Brascamp–Lieb Inequality
Coordinatewise Symmetric Log-concave Distributions
Remarks
Logarithmic Sobolev Inequalities
The Entropy Functional and Relative Entropy
Definitions and Examples
Exponential Bounds
Bounds Involving Relative Entropy
Orlicz Norms and Growth of 𝑳𝒑-norms
Bounds Involving Second Order Derivatives
Remarks
Supremum and Infimum Convolutions
Regularity and Analytic Properties
Generators
Hamilton–Jacobi Equations
Supremum/Infimum Convolution Inequalities
Transport-Entropy Inequalities
Remarks
Analysis on the Sphere
Sobolev-type Inequalities
Spherical Derivatives
Second Order Modulus of Gradient
Spherical Laplacian
Poincaré and Logarithmic Sobolev Inequalities
Isoperimetric and Cheeger-type Inequalities
Remarks
Second Order Spherical Concentration
Second Order Poincaré-type Inequalities
Bounds on the 𝑳2-norm in the Euclidean Setup
First Order Concentration Inequalities
Second Order Concentration
Second Order Concentration With Linear Parts
Deviations for Some Elementary Polynomials
Polynomials of Fourth Degree
Large Deviations for Weighted ℓ𝒑-norms
Remarks
Linear Functionals on the Sphere
First Order Normal Approximation
Second Order Approximation
Characteristic Function of the First Coordinate
Upper Bounds on the Characteristic Function
Polynomial Decay at Infinity
Remarks
First Applications to Randomized Sums
Typical Distributions
Concentration Problems for Weighted Sums
The Structure of Typical Distributions
Normal Approximation for Gaussian Mixtures
Approximation in Total Variation
𝑳𝒑-distances to the Normal Law
Lower Bounds
Remarks
Characteristic Functions of Weighted Sums
Upper Bounds on Characteristic Functions
Concentration Functions of Weighted Sums
Deviations of Characteristic Functions
Deviations in the Symmetric Case
Deviations in the Non-symmetric Case
The Linear Part of Characteristic Functions
Remarks
Fluctuations of Distributions
The Kantorovich Transport Distance
Large Deviations for the Kantorovich Distance
Pointwise Fluctuations
The Lévy Distance
Berry–Esseen-type Bounds
Preliminary Bounds on the Kolmogorov Distance
BoundsWith a Standard Rate
Deviation Bounds for the Kolmogorov Distance
The Log-concave Case
Remarks
Refined Bounds and Rates
𝑳2 Expansions and Estimates
General Approximations
Bounds for 𝑳2-distance With a Standard Rate
Expansion With Error of Order 𝒏−1
Two-sided Bounds
Asymptotic Formulas in the General Case
General Lower Bounds
Refinements for the Kolmogorov Distance
Preliminaries
Large Interval. Final Upper Bound
Relations Between Kantorovich, 𝑳2 and Kolmogorov distances
Lower Bounds
Remarks
Applications of the Second Order Correlation Condition
Mean Value of 𝝆(𝑭𝜽 ,𝚽) Under the Symmetry Assumption
Berry–Esseen Bounds Involving 𝚲
Deviations Under Moment Conditions
The Case of Non-symmetric Distributions
The Mean Value of 𝝆(𝑭𝜽 ,𝚽) in the Presence of Poincaré Inequalities
Deviations of of 𝝆(𝑭𝜽 ,𝚽) in the Presence of Poincaré Inequalities
Relation to the Thin Shell Problem
Remarks
Distributions and Coefficients of Special Type
Special Systems and Examples
Systems with Lipschitz Condition
Trigonometric Systems
Chebyshev Polynomials
Functions of the Form 𝑿𝒌 (𝒕, 𝒔) = 𝒇 (𝒌𝒕 + 𝒔)
The Walsh System on the Discrete Cube
Empirical Measures
Lacunary Systems
Remarks
Distributions With Symmetries
Coordinatewise Symmetric Distributions
Behavior On Average
Coordinatewise Symmetry and Log-concavity
Remarks
Product Measures
Edgeworth Expansion for Weighted Sums
Approximation of Characteristic Functions of Weighted Sums
Integral Bounds on Characteristic Functions
Approximation in the Kolmogorov Distance
Normal Approximation Under the 4-th Moment Condition
Approximation With Rate 𝒏−3/2
Lower Bounds
Remarks
Product Measures
Bernoulli Coefficients
Random Sums
Existence of Infinite Subsequences of Indexes
Selection of Indexes from Integer Intervals
References
Glossary
Index