Inequalities In Analysis And Probability

by Odile Pons

Overview

The book is aimed at graduate students and researchers with basic knowledge of Probability and Integration Theory. It introduces classical inequalities in vector and functional spaces with applications to probability. It also develops new extensions of the analytical inequalities, with sharper bounds and generalizations to the sum or the supremum of random variables, to martingales and to transformed Brownian motions. The proofs of the new results are presented in great detail.

Product Details

ISBN-13:	9789814412575
Publisher:	World Scientific Publishing Company, Incorporated
Publication date:	01/22/2013
Pages:	232
Product dimensions:	6.20(w) x 9.10(h) x 0.80(d)

Table of Contents

Preface v

1 Preliminaries 1

1.1 Introduction 1

1.2 Cauchy and Hölder inequalities 2

1.3 Inequalities for transformed series and functions 6

1.4 Applications in probability 9

1.5 Hardy's inequality 13

1.6 Inequalities for discrete martingales 15

1.7 Martingales indexed by continuous parameters 19

1.8 Large deviations and exponential inequalities 23

1.9 Functional inequalities 27

1.10 Content of the book 28

2 Inequalities for Means and Integrals 31

2.1 Introduction 31

2.2 Inequalities for means in real vector spaces 31

2.3 Hölder and Hilbert inequalities 35

2.4 Generalizations of Hardy's inequality 38

2.5 Carleman's inequality and generalizations 46

2.6 Minkowski's inequality and generalizations 48

2.7 Inequalities for the Laplace transform 52

2.8 Inequalities for multivariate functions 54

3 Analytic Inequalities 59

3.1 Introduction 59

3.2 Bounds for series 61

3.3 Cauchy's inequalities and convex mappings 64

3.4 Inequalities for the mode and the median 68

3.5 Mean residual time 72

3.6 Functional equations 74

3.7 Carlson's inequality 80

3.8 Functional means 83

3.9 Young's inequalities 86

3.10 Entropy and information 88

4 Inequalities for Martingales 91

4.1 Introduction 91

4.2 Inequalities for sums of independent random variables 92

4.3 Inequalities for discrete martingales 99

4.4 Inequalities for martingales indexed by R₊ 104

4.5 Poisson processes 108

4.6 Brownian motion 111

4.7 Diffusion processes 116

4.8 Level crossing probabilities 120

4.9 Martingales in the plane 124

5 Functional Inequalities 127

5.1 Introduction 127

5.2 Exponential inequalities for functional empirical processes 128

5.3 Exponential inequalities for functional martingales 135

5.4 Weak convergence of functional processes 139

5.5 Differentiable functionals of empirical processes 142

5.6 Regression functions and biased length 146

5.7 Regression functions for processes 151

6 Inequalities for Processes 153

6.1 Introduction 153

6.2 Stationary processes 154

6.3 Ruin models 156

6.4 Comparison of models 162

6.5 Moments of the processes at T_a 164

6.6 Empirical process in mixture distributions 166

6.7 Integral inequalities in the plane 169

6.8 Spatial point processes 170

7 Inequalities in Complex Spaces 179

7.1 Introduction 179

7.2 Polynomials 182

7.3 Fourier and Hermite transforms 183

7.4 Inequalities for the transforms 190

7.5 Inequalities in C 192

7.6 Complex spaces of higher dimensions 193

7.7 Stochastic integrals 197

Appendix A Probability 201

A.1 Definitions and convergences in probability spaces 201

A.2 Boundary-crossing probabilities 206

A.3 Distances between probabilities 207

A.4 Expansions in L₂(R) 210

Bibliography 213

Index 219