Normal Approximation: New Results, Methods and Problems

✍ Scribed by Vladimir V. Senatov

• Publisher: De Gruyter
• Year: 2011
• Tongue: English
• Leaves: 376
• Category: Library

✦ Table of Contents

Foreword
Preface
1 Introduction
1.1 Formulation of the problem
1.2 Historical aspects; Qualitative effects of transition from the one-dimensional case to multidimensional spaces
1.3 On the contents of the book
1.4 Some notation and definitions
2 Elements of the theory of probability metrics
2.1 Basic probability metrics
2.2 On distinctions between probability metrics
2.3 Special properties of probability metrics
2.4 Relations between metrics
2.5 Structure of the set of metrics
2.6 Estimates in equivalent metrics
2.7 The Lévy-Prokhorov metrics; Weak convergence
2.8 Uniform metrics
2.9 Metrics λ and χ
2.10 Ideal metrics
2.11 Extremal properties of L and ρ metrics
3 Method of characteristic functions; Berry-Esseen theorem
3.1 Basic properties of characteristic functions
3.2 Estimation of closeness of characteristic functions in CLT
3.3 Berry-Esseen inequality
3.4 Berry-Esseen theorem
3.5 Modifications of Berry-Esseen theorem
3.6 One remark about the method of characteristic functions in the multidimensional case
4 Method of compositions in the one-dimensional case
4.1 Preliminary remarks
4.2 Smoothing inequality
4.3 Basic estimate
4.4 Speculations
4.5 One modification of the method of compositions
4.6 Convergence rate estimate under the condition β2+δ < ∞, 0<δ<1
5 Method of compositions in the multidimensional case
5.1 Base estimates
5.2 Estimates under the condition Var(Pn, Φ) →0
6 Convergence rate estimates in weak metrics
6.1 On a relation between estimates of the Lévy-Prokhorov metrics
6.2 Estimation of the metrics π through the metrics ζ
6.3 Estimation of π(Pn, Φ; G) in the space l2
6.4 Estimation of π(Ρn, Φ; G) and π(Ρn, Φ; B) in the space Ek
6.5 Estimation of π(Ρn, Φ; B) in the space l2
6.6 A remark on the estimates in the central limit theorem for weak metrics
7 Estimation of uniform distances in l2
7.1 Some remarks on the method to prove the estimates
7.2 Additional notation
7.3 Formulation and proof of the theorem
8 Estimation of derivatives and measures
8.1 Derivatives of the functions ξ(rk(x) — ρ)/ρ) and η(rk(x) - ρ)/ρ)
8.2 Derivatives of the normal law
8.3 Estimates of the Pn-measure of the sets F(ε) and Φ(ε)
8.4 On a bound in the space Ek
9 Lower bounds for uniform metrics
Conclusion
Bibliography
Index