The book is organized according to the main concepts and topics in this area and it provides new inequalities for sums of random variables, their maximum, martingales, Brownian motions, diffusion processes, point processes and their maximum.

The emphasis on the inequalities is aimed at graduate students and researchers having the basic knowledge in Analysis, Integration Theory and Probability. The book gives a survey of classical inequalities in several fields with the main ideas of their proofs, as well as an account of inequalities in vector and functional spaces with applications to probability and to complete and extend on them. The inequalities presented are not an exhaustive list of them.

This book contains many proofs about basic inequalities for martingales with discrete or continuous parameters, and the proofs of the new results are detailed, accessible to the readers, and finally are presented in great detail. All illustrated by applications in probability theory. Its scope of study includes inequalities in vector spaces and functional Hilbert spaces in order to simplify the approach of uniform bounds for stochastic processes in functional classes. It also develops new extensions of the analytical inequalities, with sharper bounds and generalization to the sum on the maximum of random variables, to martingales and the transformed Brownian motions.

This book is an attractive introduction and extension to applications of the analytic inequalities to probability, random variables, martingales, stochastic processes with values in Banach spaces, complex spaces, tail behavior of processes, and so on.

The book helps to give to readers, to graduate students and beginners and senior researchers, a fundamental background and compendium in probability, stochastic processes, martingales, Brownian motions, and integration theory, with widely applications. I feel sure that it will be of great use both to graduate students, and researchers, in analysis and probability theory.

The book contains preface, table of contents, list de figures (p. i-ix), seven chapters, and an appendix A with the basics used in Probability Theory, as definitions and convergences in probability spaces; boundary-crossing probabilities; distances between distributions, and expansion in L2 ( R) an interesting updated bibliography with 110 references, and index, in 221 p.

The first chapter, preliminaries, (p. 1-30), introduces the basic, definitions and notions, devoted to present the origin of the inequalities for convex functions and its generalizations to inequalities for the tail distribution of sums of independent or dependent variables, and to martingales indexed by discrete or continuous sets. These inequalities are the decisive arguments for bounding series, integrals, and for proving other inequalities.

The chapter introduces the following items: Cauchy and Hölder inequalities; inequalities for transformed series and functions; applications in Probability Theory; Hardy´s inequality; inequalities for discrete martingales; martingales indexed by continuous parameters; large deviations and exponential inequalities, and finally functional inequalities. Five remarkable theorems and two propositions are presented here.

The next chapters develop extension and applications of the classical results presented in above chapter.

In the chapter two, inequalities for means and integrals, (p. 31-58), on extends, the Cauchy, Hölder and Hilbert inequalities for arithmetic and integral means in real spaces. Hardy´s inequality is presented with new versions of weighted inequalities in real analysis. This chapter deals basically with 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, and inequalities for multivariate functions.

This chapter works on ten theorems, nineteen propositions and eight examples that illustrating the different theoretical lines proposed. Some are quite difficult for the beginner.

The chapter three, (p.59-90), analytic inequalities, presents some of the most important analytic inequalities for the arithmetic and geometric means, in particular functional means for the power and the logarithm functions. The upper and lower bounds for the logarithm mean function on R+, Carlson´s inequality, is improved and is applied to other functions. On the other hand, inequalities for the arithmetic and the geometric means extend Cauchy´s results, and also established results for the median, the mode, mean residual time, and the mean of density or distributions functions. Finally, functional equations, Young´s integral inequality and several results about the entropy are proved.

The following subjects are considered: 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, and th3e concepts of entropy and information.

Three theorems, twenty six propositions and six examples, with their respective proofs are deals with great detail, and it shows the excellent work on them.

In the chapter four, (p. 91-126), inequalities for martingales, on deal inequalities for sums and maximum of n independent random variables, extended to discrete and continuous martingales an to their maximum variables. The Bürkholder- Davis-Gundy inequalities are improving. Moreover, the Chernov and Bennet theorems and other exponential inequalities are generalized to local martingales, applied to Brownian motions and Poisson processes, and turn out generalized in several forms to dependent and bounded variables and to local martingales, using Lenglart´s inequality. Finally another question is also considered, the solutions of diffusion equations are explicitly established, and then, some level crossing problems are studied.

This chapter presents the following statements inequalities for sums of independent random variables; inequalities for discrete martingales and for martingales indexed by R+; Poisson processes; Brownian motion; Diffusion processes; Level crossing probabilities, and martingales in the plane.

The chapter is illustrated with nine theorems, three lemmas, one corollary, and twenty four propositions elaborated, all them intensively and adequately proved.

The chapter five, Functional inequalities, (p. 127-152), concerns inequalities in functional spaces, for sums of real functions of random variables and their supremum on class of the different kinds defining the transformed variables. Functional of discrete or continuous martingales are extended, using the Chernov´s, Hoeffding´s and Bennet´s theorems. Uniform versions of the Bürkholder- Davis-Gundy inequalities, allows extended the above functional of martingales. Moreover some applications to the weak convergence are considered.

Chapter 5 analyzed exponential inequalities for functional empirical processes; exponential inequalities for functional martingales; weak convergence of functional processes; differentiable functional of empirical processes; regression functions and biased length, and finally regression functions for processes.

On presented here six very interesting theorems, one lemma, three corollaries, sixteen propositions and two examples, for illustrating the mainly concepts in this chapter.

The chapter six, inequalities for processes, (p. 153- 178), introduces a very useful perspective about inequalities for stochastic processes, in particular, new results for Gaussian processes, and the distribution of the ruin time of the Sparre Anderson ruin model and some more optimistic stochastic models with a diffusion term. By end, some spatial stationary measures, and also their weak convergence deduced from their tail behavior.

Chapter 6, concerns stationary processes; ruin models; comparison of models; moments of the processes at Tu; empirical process in mixture distributions; integral inequalities in the plane and spatial point processes.

The chapter is illustrated with four theorems, five lemmas, one corollary, and seven propositions elaborated, all them intensively and adequately proved, some are a little hard.

The chapter seven, Inequalities in complex spaces, (p. 179-200), focuses on complex spaces and on the Fourier transform. The classical theory is extended to generalize the expansions of analytic functions of several variables in series. On study the orthonormal basis of the Hermite polynomials and their properties, with the orders of the approximations. The isometry between R2 and C, extended to the same between R3 and a complex space where the Fourier transform is defined. Conditions for the differentiability of complex functions, as Cauchy conditions, are established, as well as in higher dimensions.

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Chapter 7 presents some interesting questions as polynomials; Fourier and Hermite transforms; inequalities for the transforms; inequalities in C; complex spaces of higher dimensions and stochastic integrals.

The chapter included one special theorem, one lemma, two corollaries, eight propositions and two examples, with their proofs. A great detailed study about above them, are done.

An appendix A closed the book, with the basics of probability, definitions kinds or convergences in probability spaces, distances between probabilities, and other questions basic of interest.

In detailed form:

Appendix A. Probability (p. 201)

1. Definitions and convergences in probability spaces (p. 201-206).

2. Boundary-crossing probabilities (p. 206-207)

3. Distances between probabilities (p.207-209)

4. Expansions in