This book is an introduction to probability theory covering laws of large numbers, central limit theorems, random walks, martingales, Markov chains, ergodic theorems, and Brownian motion. It is a comprehensive treatment concentrating on the results that are the most useful for applications. Its philosophy is that the best way to learn probability is to see it in action, so there are 200 examples and 450 problems.

Probability: Theory and Examples

* Measure Theory-Basic Notions * Measure Theory-Key Results * Processes, Distributions, and Independence * Random Sequences, Series, and Averages * Characteristic Functions and Classical Limit Theorems * Conditioning and Disintegration * Martingales and Optional Times * Markov Processes and Discrete-Time Chains * Random Walks and Renewal Theory * Stationary Processes and Ergodic Theory * Special Notions of Symmetry and Invariance * Poisson and Pure Jump-Type Markov Processes * Gaussian Processes and Brownian Motion * Skorohod Embedding and Invariance Principles * Independent Increments and Infinite Divisibility * Convergence of Random Processes, Measures, and Sets * Stochastic Integrals and Quadratic Variation * Continuous Martingales and Brownian Motion * Feller Processes and Semigroups * Ergodic Properties of Markov Processes * Stochastic Differential Equations and Martingale Problems * Local Time, Excursions, and Additive Functionals * One-Dimensional SDEs and Diffusions * Connections with PDEs and Potential Theory * Predictability, Compensation, and Excessive Functions * Semimartingales and General Stochastic Integration * Large Deviations * Appendix 1: Advanced Measure Theory * Appendix 2: Some Special Spaces * Historical and Bibliographical Notes * Bibliography * Indices

http://cyber.sibsutis.ru:82/Monarev/docs/nauka/PROBABILITY/Kallenberg%20O.%20Foundations%20of%20Modern%20Probability%20(Springer,1997)(ISBN%200387949577)(535s).pdf

Foundations of modern probability

1. Basic convexity 2. Boundary structure 3. Minkowski addition 4. Curvature measure and quermass integrals 5. Mixed volumes 6. Inequalities for mixed volumes 7. Selected applications Appendix.

/pdf/convex-bodies-the-brunn-minkowski-theory-3raynvod45.pdf

Convex bodies : the Brunn-Minkowski theory

This book was first published in 1943 and then was reprinted several times with corrections. It presents the development of the classical problem of moments for the first 50 years, after its introduction by Stieltjes in the 1890s. In addition to initial developments by Stieltjes, Markov, and Chebyshev, later contributions by Hamburger, Nevanlinna, Hausdorff, Stone, and others are discussed. The book also contains some results on the trigonometric moment problem and a chapter devoted to approximate quadrature formulas.

The problem of moments

Publisher Summary The fundamental assumption of all theories of plasticity—that of time independence of the equations of state—makes simultaneous description of the plastic and rheologic properties of a material impossible. I t is well-known that in many practical problems, the actual behaviour of a material is governed by plastic as well as by rheologic effects. It can even be said that for many important structural materials, rheologic effects are more pronounced after the plastic state has been reached. Every material shows more or less pronounced viscous properties. In some problems, the influence of viscous properties of the material may be negligible, while in others, it may be essential. Both sciences—plasticity and rheology—are concerned with the description of important mechanical properties of structural materials. Each of them has created its own methods of investigation and has developed within the framework of certain assumptions which, unfortunately, cannot always be satisfied in reality. The results of rheology are confined to cases where plastic strain is of no decisive importance.

Fundamental Problems in Viscoplasticity

On the Law of the Iterated Logarithm

In this paper, the analogues of the theorems of [4], [5 1 oIn solutions of elliptic partial differential equations will be obtained for the case where the lnumber of independent variables exceeds 2. For the sake of notational simplicity, it will be assumed that the number of independent variables is 3. The equation to be considered is of the type Au + 0, where Au is the Euclidean Laplacian of i, and no second order partial derivative of i occurs in the rest of the equation. The replacement of Au by a more general linear combinationi of second derivatives of u will not be considered at this time. (In the plane, this more general case can be reduced to the special case by colnformal mappings under suitable smoothness assumptions on the coefficients; in space, perturbatioin imiethods of Korn and Lichtenstein can be used.) For simplicitv, the partial differential equation to be considered will be assumed to be linear (the methods are applicable to non-linear equations of the type (5)-(6) below). The first part of the paper deals with solutions near a zero, the second part with solutions near an isolated singularity.

On the Local Behavior of Solutions of Non-Parabolic Partial Differential Equations

http://www.jstor.org/stable/pdfplus/2372182.pdf

On the non-existence of conjugate points.*

(2) t~~~~~~~ (0) Y , fn e*. n=-co Let f (f-, f+), where f+ and f- are the respective sequences f+ = (fo, f,** f- = * , f-2., f4)., and let the corresponding functions f+ ((p) i f- (p) of class L2 be called the interior and exterior functions of f (Q), 00 oo

The spectra of toeplitz's matrices.*

where ,, = Im, y(t) = ym(t) and m 1, , d. This was proved by Perron (cf. [8], pp. 158-159) with the lim replaced by lim sup, but his method can be used to supply (iv) as it stands, cf. [7]. In view of known results ([2], [3]) for the case when (i) is equivalent to a second order differential equation, it is natural to ask under what conditions, weaker than (iii), will the assertion concerning (iv) remain true. In this direction, it will be shown that (iii) can be relaxed to

http://www.jstor.org/stable/pdfplus/2372422.pdf

Aurel Wintner

Papers

On the Law of the Iterated Logarithm

On the Local Behavior of Solutions of Non-Parabolic Partial Differential Equations

On the non-existence of conjugate points.*

The spectra of toeplitz's matrices.*

Asymptotic Integrations of Linear Differential Equations