Inequalities in Analysis and Probability

The following sections are included:

• Preface

• Contents

The following sections are included:

• Introduction

• Cauchy and Hölder inequalities

• Inequalities for transformed series and functions

• Applications in probability

• Hardy’s inequality

• Inequalities for discrete martingales

• Martingales indexed by continuous parameters

• Large deviations and exponential inequalities

• Functional inequalities

• Content of the book

The following sections are included:

• Introduction

• Inequalities for means in real vector spaces

• Hölder and Hilbert inequalities

• Generalizations of Hardy’s inequality

• Carleman’s inequality and generalizations

• Minkowski’s inequality and generalizations

• Inequalities for the Laplace transform

• Inequalities for multivariate functions

The following sections are included:

• Introduction

• Bounds for series

• Cauchy’s inequalities and convex mappings

• Inequalities for the mode and the median

• Mean residual time

• Functional equations

• Carlson’s inequality

• Functional means

• Young’s inequalities

• Entropy and information

The following sections are included:

• Introduction

• Inequalities for sums of independent random variables

• Inequalities for discrete martingales

• Inequalities for martingales indexed by ℝ₊

• Poisson processes

• Brownian motion

• Diffusion processes

• Level crossing probabilities

• Martingales in the plane

The following sections are included:

• Introduction

• Exponential inequalities for functional empirical processes

• Exponential inequalities for functional martingales

• Weak convergence of functional processes

• Differentiable functionals of empirical processes

• Regression functions and biased length

• Regression functions for processes

The following sections are included:

• Introduction

• Stationary processes

• Ruin models

• Comparison of models

• Moments of the processes at T_a

• Empirical process in mixture distributions

• Integral inequalities in the plane

• Spatial point processes

The following sections are included:

• Introduction

• Polynomials

• Fourier and Hermite transforms

• Inequalities for the transforms

• Inequalities in ℂ

• Complex spaces of higher dimensions

• Stochastic integrals

The following sections are included:

• Appendix A: Probability
  • Definitions and convergences in probability spaces
  • Boundary-crossing probabilities
  • Distances between probabilities
  • Expansions in L₂(ℝ)

• Bibliography

• Index