Inequalities In Analysis And Probability (Third Edition)

The book introduces classical inequalities in vector and functional spaces with applications to probability. It develops new analytical inequalities, with sharper bounds and generalizations to the sum or the supremum of random variables, to martingales, to transformed Brownian motions and diffusions, to Markov and point processes, renewal, branching and shock processes. In this third edition, the inequalities for martingales are presented in two chapters for discrete and time-continuous local martingales with new results for the bound of the norms of a martingale by the norms of the predictable processes of its quadratic variations, for the norms of their supremum and their p-variations. More inequalities are also covered for the tail probabilities of Gaussian processes and for spatial processes. This book is well-suited for undergraduate and graduate students as well as researchers in theoretical and applied mathematics. Contents Preface 1. Preliminaries 1.1 Introduction 1.2 Cauchy and Hölder inequalities 1.3 Inequalities for transformed series and functions 1.4 Applications in probability 1.5 Hardy's inequality 1.6 Inequalities for discrete martingales 1.7 Martingales indexed by continuous parameters 1.8 Large deviations and exponential inequalities 1.9 Functional inequalities 1.10 Content of the book 2. Inequalities for Means and Integrals 2.1 Introduction 2.2 Inequalities for means in real vector spaces 2.3 HŁolder and Hilbert inequalities 2.4 Generalizations of Hardy's inequality 2.5 Carleman's inequality and generalizations 2.6 Minkowski's inequality and generalizations 2.7 Inequalities for the Laplace transform 2.8 Inequalities for multivariate functions 3. Analytic Inequalities 3.1 Introduction 3.2 Bounds for series 3.3 Cauchy's inequalities and convex mappings 3.4 Inequalities for the mode and the median 3.5 Mean residual time 3.6 Functional equations 3.7 Carlson's inequality 3.8 Functional means 3.9 Young's inequalities 3.10 Entropy and information 4. Inequalities for Discrete Martingales 4.1 Introduction 4.2 Inequalities for sums of independent random variables 4.3 Inequalities for discrete martingales 4.4 Inequalities for first passage and maximum 4.5 Inequalities for p-order variations 4.6 Weak convergence of discrete martingales 5. Inequalities for Time-Continuous Martingales 5.1 Introduction 5.2 Inequalities for martingales indexed by R+ 5.3 Inequalities for the maximum 5.4 Inequalities for p-order variations 5.5 Weak convergence of martingales and point processes 5.6 Poisson and renewal processes 5.7 Brownian motion 5.8 Diffusion processes 5.9 Martingales in the plane 6. Stochastic Calculus 6.1 Stochastic integration 6.2 Exponential solutions of differential equations 6.3 Exponential martingales, submartingales 6.4 Gaussian processes 6.5 Processes with independent increments 6.6 Semi-martingales 6.7 Level crossing probabilities 6.8 Local times 7. Functional Inequalities 7.1 Introduction 7.2 Exponential inequalities for functional empirical processes 7.3 Inequalities for functional martingales 7.4 Weak convergence of functional processes 7.5 Differentiable functionals of empirical processes 7.6 Regression functions and biased length 7.7 Regression functions for processes 7.8 Functional inequalities and applications 8. Markov Processes 8.1 Ergodic theorems 8.2 Inequalities for Markov processes 8.3 Convergence of diffusion processes 8.4 Branching process 8.5 Renewal processes 8.6 Maximum variables 8.7 Shock process 8.8 Laplace transform 8.9 Time-space Markov processes 9. Inequalities for Processes 9.1 Introduction 9.2 Stationary processes 9.3 Ruin models 9.4 Comparison of models 9.5 Moments of the processes at Ta 9.6 Empirical process in mixture distributions 9.7 Integral inequalities in the plane 9.8 Spatial point processes 9.9 Spatial Gaussian processes 10. Inequalities in Complex Spaces 10.1 Introduction 10.2 Polynomials 10.3 Fourier and Hermite transforms 10.4 Inequalities for the transforms 10.5 Inequalities in C 10.6 Complex spaces of higher dimensions 10.7 Stochastic integrals Appendix A Probability A.1 Definitions and convergences in probability spaces A.2 Boundary-crossing probabilities A.3 Distances between probabilities A.4 Expansions in L2(R) Bibliography Index "The book is complementary to the classical courses on vector and functional Hilbert spaces, integration theory and probability which focus on inequalities; The preliminary chapter introduces most classical inequalities which are generalized to more complex models with detailed proofs; It covers different domains from existing books and contains many new results; It gathers inequalities from several domains of mathematics which are not generally presented together, in a unified approach; The topics include many stochastic processes with specific inequalities and the basis of the stochastic calculus is developed in numerous applications"-- Provided by publisher 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.