• Preface--
• 1. Introduction--
• 2. Events and probabilities--
• 3. Random variables, means and variances--
• 4. Conditioning and independence--
• 5. Generating functions and the central limit theorem--
• 6. Confidence intervals for 1-parameter models--
• 7. Conditional pdfs and multi-parameter Bayesian statistics--
• 8. Linear models, ANOVA etc--
• 9. Some further probability--
• 10. Quantum probability and quantum computing-- Appendix A. Some prerequisites and addenda-- Appendix B. Discussion of some selected exercises-- Appendix C. Tables-- Appendix D. A small sample of the literature-- Bibliography-- Index.
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Statistics do not lie, nor is probability paradoxical. You just have to have the right intuition. In this lively look at both subjects, David Williams convinces mathematics students of the intrinsic interest of Statistics and Probability, and Statistics students that the language of mathematics can bring real insight and clarity to their subject. He helps students build the intuition needed, in a presentation enriched with examples drawn from all manner of applications, e.g., genetics, filtering, the Black-Scholes option-pricing formula, quantum probability and computing, and classical and modern statistical models. Statistics chapters present both the Frequentist and Bayesian approaches, emphasising Confidence Intervals rather than Hypothesis Test, and include Gibbs-sampling techniques for the practical implementation of Bayesian methods. A central chapter gives the theory of Linear Regression and ANOVA, and explains how MCMC methods allow greater flexibility in modelling. C or WinBUGS code is provided for computational examples and simulations. Many exercises are included; hints or solutions are often provided.
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• 1. A branching-process example-- Part I. Foundations:
• 2. Measure spaces--
• 3. Events--
• 4. Random variables--
• 5. Independence--
• 6. Integration--
• 7. Expectation--
• 8. An easy strong law: product measure-- Part II. Martingale Theory:
• 9. Conditional expectation--
• 10. Martingales--
• 11. The convergence theorem--
• 12. Martingales bounded in L2--
• 13. Uniform integrability--
• 14. UI martingales--
• 15. Applications-- Part III. Characteristic Functions:
• 16. Basic properties of CFs--
• 17. Weak convergence--
• 18. The central limit theorem-- Appendices-- Exercises.
• (source: Nielsen Book Data)

Probability theory is nowadays applied in a huge variety of fields including physics, engineering, biology, economics and the social sciences. This book is a modern, lively and rigorous account which has Doob's theory of martingales in discrete time as its main theme. It proves important results such as Kolmogorov's Strong Law of Large Numbers and the Three-Series Theorem by martingale techniques, and the Central Limit Theorem via the use of characteristic functions. A distinguishing feature is its determination to keep the probability flowing at a nice tempo. It achieves this by being selective rather than encyclopaedic, presenting only what is essential to understand the fundamentals; and it assumes certain key results from measure theory in the main text. These measure-theoretic results are proved in full in appendices, so that the book is completely self-contained. The book is written for students, not for researchers, and has evolved through several years of class testing. Exercises play a vital role. Interesting and challenging problems, some with hints, consolidate what has already been learnt, and provide motivation to discover more of the subject than can be covered in a single introduction.
(source: Nielsen Book Data)

• v. 1. Foundations.--
• v. 2. Itô calculus.

• Some frequently used notation--
• 1. Brownian motion-- Part I. Introduction--
• 2. Basics about Brownian motion--
• 3. Brownian motion in higher dimensions--
• 4. Gaussian processes and Levy processes-- Part II. Some Classical Theory--
• 5. Basic measure theory--
• 6. Basic probability theory--
• 7. Stochastic processes--
• 8. Discrete-parameter martingale theory--
• 9. Continuous-parameter martingale theory--
• 10. Probability measure on Lusin spaces-- Part III. Markov Processes:
• 11. Transition functions and resolvents--
• 12. Feller-Dynkin processes--
• 13. Additive functionals--
• 14. Approach to ray processes: the Martin boundary--
• 15. Ray processes--
• 16. Applications-- References-- Index.
• (source: Nielsen Book Data)

• Some frequently used notation--
• 4. Introduction to Ito calculus-- 4.1. Some motivating remarks-- 4.2. Some fundamental ideas: previsible processes, localization, etc.-- 4.3. The elementary theory of finite-variation processes-- 4.4. Stochastic integrals: the L2 theory-- 4.5. Stochastic integrals with respect to continuous semimartingales-- 4.6. Applications of Ito's formula--
• 5. Stochastic differential equations and diffusions-- 5.1. Introduction-- 5.2. Pathwise uniqueness, strong SDEs, flows-- 5.3. Weak solutions, uniqueness in law-- 5.4. Martingale problems, Markov property-- 5.5. Overture to stochastic differential geometry-- 5.6. One-dimensional SDEs-- 5.7. One-dimensional diffusions--
• 6. The general theory-- 6.1. Orientation-- 6.2. Debut and section theorems-- 6.3. Optional projections and filtering-- 6.4. Characterising previsible times-- 6.5. Dual previsible projections-- 6.6. The Meyer decomposition theorem-- 6.7. Stochastic integration: the general case-- 6.8. Ito excursion theory-- References-- Index.
• (source: Nielsen Book Data)

Now available in paperback, this celebrated book has been prepared with readers' needs in mind, remaining a systematic guide to a large part of the modern theory of Probability, whilst retaining its vitality. The authors' aim is to present the subject of Brownian motion not as a dry part of mathematical analysis, but to convey its real meaning and fascination. The opening, heuristic chapter does just this, and it is followed by a comprehensive and self-contained account of the foundations of theory of stochastic processes. Chapter 3 is a lively and readable account of the theory of Markov processes. Together with its companion volume, this book helps equip graduate students for research into a subject of great intrinsic interest and wide application in physics, biology, engineering, finance and computer science.
(source: Nielsen Book Data)
This celebrated book has been prepared with readers' needs in mind, remaining a systematic treatment of the subject whilst retaining its vitality. The second volume follows on from the first, concentrating on stochastic integrals, stochastic differential equations, excursion theory and the general theory of processes. Much effort has gone into making these subjects as accessible as possible by providing many concrete examples that illustrate techniques of calculation, and by treating all topics from the ground up, starting from simple cases. Many of the examples and proofs are new; some important calculational techniques appeared for the first time in this book. Together with its companion volume, this book helps equip graduate students for research into a subject of great intrinsic interest and wide application in physics, biology, engineering, finance and computer science.
(source: Nielsen Book Data)

• v. 1. Foundations
• v. 2. Itô calculus

This updated introduction to the field has been completely revised to include the latest research on probability. The textbook has also been restructured to improve its logical progression and ease of use.
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• The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers - dissipation of energy in the locally isotropic turbulence, A.N. Kolmogorov-- Kolmogorov's hydrodynamic attractors, V.I. Arnol'd-- approximation of attractors, large eddy simulations and multiscale methods, R. Temam-- transition functions, paths and path integrals, D. Williams-- turbulent cascade and intermittency growth, R.H. Kraichnan-- random multifractals - negative dimensions and the resulting limitations of the thermodynamic formalism, B.B. Mandelbrot-- from global scaling, a la Kolmogorov, to local multifractal scaling in developed turbulence, U. Frisch-- structure and dynamics of homogenous turbulence models and simulations, Z-S She, E. Jackson and S.A. Orszag-- the Kolomorogov spectrum and its oceanic cousins - a review, O.M. Phillips-- local isotropy of the smallest scales of turbulent scalar and velocity fields, C. Van Atta-- Kolmogorov similarity hypotheses for scalar fields - sampling intermittent turbulent mixing in the ocean and galaxy, C.H. Gibson-- on local isotropy of passive scalars in turbulent shear flows, K.R. Sreenivasan-- Kolmogorov's contributions to the physical and geometrical understanding of small-scale turbulence and recent developments, J.C.R. Hunt and J.C. Vassilicos-- Kolmogorov's two-equation model of turbulence, D.B. Spalding-- some applications of Kolmogorov's turbulence research in the field of combustion, K.N.C. Bray and R.S. Cant.
• (source: Nielsen Book Data)

It is 50 years since the publication of A.N. Kolmogorov's paper in 1941, probably the most famous paper ever written on turbulence. This volume commemorates this and other papers which established many of the key ideas of probability theory, stochastic processes and turbulence. These have since found application in many areas of research, including the theory of the turbulent motions of fluids, diffusion theory, the theory of automata, and communication theory. In putting together this volume the editors asked their colleagues to review, in the context of recent research, Kolmogorov's different contributions. And because his original papers are so important and not widely available, translations of two of his 1941 papers and one 1942 paper on turbulence have been included. Most of the fundamental problems in turbulence being investigated today were known to Kolmogorov, and he established lines of research which are still being followed and are attracting new methods of analysis. For example, Mandelbrot and Frisch here describe the recent ideas of multi-fractals and their use in understanding turbulence. Other contributions cover probability theory, small-scale turbulence, branching processes, fractal variables, general turbulent flows and turbulent combustion.
(source: Nielsen Book Data)