Concentration and Gaussian Approximation for Randomized Sums

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