Concentration and Gaussian Approximation for Randomized Sums

Inhaltsverzeichnis

1. Frontmatter

2. Generalities

   1. Frontmatter

   2. Chapter 1. Moments and Correlation Conditions

      Sergey Bobkov, Gennadiy Chistyakov, Friedrich Götze

      Abstract

      This definition is frequently used in Convex Geometry, especially for random vectors which are uniformly distributed over a convex body (in which case the body is called isotropic, cf. [144]).

   3. Chapter 2. Some Classes of Probability Distributions

      Sergey Bobkov, Gennadiy Chistyakov, Friedrich Götze

      Abstract

      The relevance of the previously introduced moment and correlation-type functionals can be illustrated by examples of some classical classes of probability distributions. In this chapter, these functionals are discussed for product measures (in which case one can also refine upper bounds on “small ball” probabilities), for joint distributions of pairwise independent random variables, and for coordinate-symmetric distributions. We also discuss the class of logarithmically concave measures and include some additional background material which will be needed later on.

   4. Chapter 3. Characteristic Functions

      Sergey Bobkov, Gennadiy Chistyakov, Friedrich Götze

      Abstract

      The moment functionals we discussed before may be explicitly expressed in terms of characteristic functions of linear functionals of a given random vector. However, information on various bounds on characteristic functions and their deviations from the characteristic function of another law on the real line will be needed for a different purpose – to study the Kolmogorov and Lévy distances between the corresponding distribution functions. In this chapter, we describe general tools in the form of smoothing and Berry–Esseen-type inequalities, which allow one to perform the transition from the results about closeness or smallness of Fourier–Stieltjes transforms to corresponding results about the associated functions of bounded variation.

   5. Chapter 4. Sums of Independent Random Variables

      Sergey Bobkov, Gennadiy Chistyakov, Friedrich Götze

      Abstract

      In this chapter we collect basic definitions and results on the normal approximation of distributions of independent random variables (in Kolmogorov distance), and also discuss possible improved rates of approximation when replacing the normal law by corresponding Edgeworth corrections. The first section deals with moment based quantities for single random variables

3. Selected Topics on Concentration

   1. Frontmatter

   2. Chapter 5. Standard Analytic Conditions

      Sergey Bobkov, Gennadiy Chistyakov, Friedrich Götze

      Abstract

      In some problems/Sobolev-type inequalities, it makes sense to slightly modify the notion of the generalized modulus of gradient.

   3. Chapter 6. Poincaré-type Inequalities

      Sergey Bobkov, Gennadiy Chistyakov, Friedrich Götze

      Abstract

      In connection with the isoperimetric problem in metric probability spaces, Poincarétype inequalities have been already discussed in Chapter 5, where we also described basic examples. Such inequalities also serve as the most convenient route to concentration results.

   4. Chapter 7. Logarithmic Sobolev Inequalities

      Sergey Bobkov, Gennadiy Chistyakov, Friedrich Götze

      Abstract

      Logarithmic Sobolev inequalities strengthen Poincaré-type inequalities, which allows one to derive sharper deviation inequalities for various classes of functions, not necessarily under the Lipschitz hypothesis. To introduce this class of analytic inequalities, first we briefly mention basic properties of the involved entropy functional and then describe several important examples of measures satisfying logarithmic Sobolev inequalities. The remaining part of the chapter deals with various bounds that are valid in the presence of logarithmic Sobolev inequalities.

   5. Chapter 8. Supremum and Infimum Convolutions

      Sergey Bobkov, Gennadiy Chistyakov, Friedrich Götze

      Abstract

      Logarithmic Sobolev inequalities are closely related to another important class of inequalities in terms of the so-called supremum- and infimum-convolutions, whose advantage is that they do not require smoothness or even continuity of the functions. It is therefore not surprising that supremum- and infimum-convolution inequalities find a wide range of applications.

4. Analysis on the Sphere

   1. Frontmatter

   2. Chapter 9. Sobolev-type Inequalities

      Sergey Bobkov, Gennadiy Chistyakov, Friedrich Götze

      Abstract

      According to the general equation (5.4), and since the geodesic and Euclidean distances are infinitesimally equivalent, the second order modulus of the gradient for functions f on the unit sphere is defined by

   3. Chapter 10. Second Order Spherical Concentration

      Sergey Bobkov, Gennadiy Chistyakov, Friedrich Götze

      Abstract

      In this chapter we discuss natural conditions imposed on the functions f on the unit sphere which guarantee smaller deviations of f from their means with respect to growing dimension n in comparison with deviations that are valid for the entire class of Lipschitz functions. These conditions involve derivatives of f of the second order, which may be considered both in the spherical and Euclidean setup.

   4. Chapter 11. Linear Functionals on the Sphere

      Sergey Bobkov, Gennadiy Chistyakov, Friedrich Götze

      Abstract

      The aim is in particular to quantify the asymptotic normality of these distributions and to include dimensional refinements of such approximation in analogy with Edgeworth expansions (which however we consider up to order 2).

5. First Applications to Randomized Sums

   1. Frontmatter

   2. Chapter 12. Typical Distributions

      Sergey Bobkov, Gennadiy Chistyakov, Friedrich Götze

      Abstract

      In this part (Chapters 12–14) we describe various aspects of Sudakov’s theorem.We start with the so-called typical distributions in his theorem and compare them with the standard normal law by means of the variance-type functionals

   3. Chapter 13. Characteristic Functions of Weighted Sums

      Sergey Bobkov, Gennadiy Chistyakov, Friedrich Götze

      Abstract

      In order to study deviations of the distribution functions F_𝜃 from the typical distribution F by means of the Kolmogorov distance, Berry–Esseen-type inequalities, which we discussed in Chapter 3, will be used. To this end we need to focus first on the behavior of characteristic functions of F_𝜃.

   4. Chapter 14. Fluctuations of Distributions

      Sergey Bobkov, Gennadiy Chistyakov, Friedrich Götze

      Abstract

      In order to deal with the main Problem 12.1.2, we start with the Kantorovich distance for bounding possible fluctuations of F_𝜃 around F on average.

6. Refined Bounds and Rates

   1. Frontmatter

   2. Chapter 15. L2 Expansions and Estimates

      Sergey Bobkov, Gennadiy Chistyakov, Friedrich Götze

      Abstract

      We now consider more precise assertions about fluctuations of the distribution functions F_𝜃 (x) of the weighted sums X, 𝜃around their means F(x).

   3. Chapter 16. Refinements for the Kolmogorov Distance

      Sergey Bobkov, Gennadiy Chistyakov, Friedrich Götze

      Abstract

      Let us now explain how this upper bound can be used to refine the lower bound (16.3). The argument is based on the following general elementary observation.

   4. Chapter 17. Applications of the Second Order Correlation Condition

      Sergey Bobkov, Gennadiy Chistyakov, Friedrich Götze

      Abstract

      We continue to use the notations of the previous chapters and denote by F_𝜃 the distribution (function) of the linear form

7. Distributions and Coefficients of Special Type

   1. Frontmatter

   2. Chapter 18. Special Systems and Examples

      Sergey Bobkov, Gennadiy Chistyakov, Friedrich Götze

      Abstract

      We also get an analogous pointwise lower bound on the “essential” part of the unit sphere. It seems however natural that the logarithmic factor could be removed from the left-hand side. A similar statement is also true for the Kolmogorov distance.

   3. Chapter 19. Distributions With Symmetries

      Sergey Bobkov, Gennadiy Chistyakov, Friedrich Götze

      Abstract

      Let us take expectations of both sides with respect to the 𝑋_𝑘’s and then insert the expectation on the left-hand side inside the supremum.

   4. Chapter 20. Product Measures

      Sergey Bobkov, Gennadiy Chistyakov, Friedrich Götze

      Abstract

      In this chapter we shall discuss the classical scheme of sums of independent random variables, which will allow us to sharpen many of the previous results. In particular, the logarithmic factor appearing in the bounds for the Kolmogorov distance in Propositions 17.1.1 and 17.5.1 may be removed (as well as in the deviation bound of Proposition 17.6.1). This is shown using Fourier Analysis, more precisely – a third order Edgeworth expansion for characteristic functions under the 4-th moment condition (cf. Chapter 4), and applying several results from Chapter 10 about deviations of elementary polynomials on the unit sphere. Even better bounds hold when applying a fourth order Edgeworth expansion under the 5-th moment condition.

   5. Chapter 21. Coefficients of Special Type

      Sergey Bobkov, Gennadiy Chistyakov, Friedrich Götze

      Abstract

      This statement may be viewed as a discrete analog of Propositions 14.4.1–14.4.2 for general spherical coefficients.

8. Backmatter