Probability and Random Processes

This self-contained, practical, entry-level text integrates the basic principles of applied mathematics, applied probability, and computational science in a presentation of stochastic processes and co

First-passage properties underlie a wide range of stochastic processes, such as diffusion-limited growth, neuron firing and the triggering of stock options. This book provides a unified presentation of first-passage processes, which highlights its interrelations with electrostatics and the resulting powerful consequences. The author begins with a presentation of fundamental theory including the connection between the occupation and first-passage probabilities of a random walk, and the connection to electrostatics and current flows in resistor networks. The consequences of this theory are then developed for simple, illustrative geometries including the finite and semi-infinite intervals, fractal networks, spherical geometries and the wedge. Various applications are presented including neuron dynamics, self-organized criticality, diffusion-limited aggregation, the dynamics of spin systems and the kinetics of diffusion-controlled reactions. First-passage processes provide an appealing way

Free Probability Theory studies a special class of 'noncommutative'random variables, which appear in the context of operators on Hilbert spaces and in one of the large random matrices. Since its emergence in the 1980s, free probability has evolved into an established field of mathematics with strong connections to other mathematical areas, such as operator algebras, classical probability theory, random matrices, combinatorics, representation theory of symmetric groups. Free probability also connects to more applied scientific fields, such as wireless communication in electrical engineering. This 2006 book gives a self-contained and comprehensive introduction to free probability theory which has its main focus on the combinatorial aspects. The volume is designed so that it can be used as a text for an introductory course (on an advanced undergraduate or beginning graduate level), and is also well-suited for the individual study of free probability.

Random walks have proven to be a useful model in understanding processes across a wide spectrum of scientific disciplines. Elements of the Random Walk is an introduction to some of the most powerful and general techniques used in the application of these ideas. The mathematical construct that runs through the analysis of the topics covered in this book, unifying the mathematical treatment, is the generating function. Although the reader is introduced to analytical tools, such as path-integrals and field-theoretical formalism, the book is self-contained in that basic concepts are developed and relevant fundamental findings fully discussed. Mathematical background is provided in supplements at the end of each chapter, when appropriate. This text will appeal to graduate students across science, engineering and mathematics who need to understand the applications of random walk techniques, as well as to established researchers.

Point processes and random measures find wide applicability in telecommunications, earthquakes, image analysis, spatial point patterns, and stereology, to name but a few areas. The authors have made a

Describes the leading techniques for analyzing noise.Discusses methods that are applicable to periodic signals, aperiodic signals, or random processes over finite or infinite intervals.Provides reader

Rigorous probabilistic arguments, built on the foundation of measure theory introduced eighty years ago by Kolmogorov, have invaded many fields. Students of statistics, biostatistics, econometrics, finance, and other changing disciplines now find themselves needing to absorb theory beyond what they might have learned in the typical undergraduate, calculus-based probability course. This 2002 book grew from a one-semester course offered for many years to a mixed audience of graduate and undergraduate students who have not had the luxury of taking a course in measure theory. The core of the book covers the basic topics of independence, conditioning, martingales, convergence in distribution, and Fourier transforms. In addition there are numerous sections treating topics traditionally thought of as more advanced, such as coupling and the KMT strong approximation, option pricing via the equivalent martingale measure, and the isoperimetric inequality for Gaussian processes. The book is not ju

Rigorous probabilistic arguments, built on the foundation of measure theory introduced eighty years ago by Kolmogorov, have invaded many fields. Students of statistics, biostatistics, econometrics, finance, and other changing disciplines now find themselves needing to absorb theory beyond what they might have learned in the typical undergraduate, calculus-based probability course. This 2002 book grew from a one-semester course offered for many years to a mixed audience of graduate and undergraduate students who have not had the luxury of taking a course in measure theory. The core of the book covers the basic topics of independence, conditioning, martingales, convergence in distribution, and Fourier transforms. In addition there are numerous sections treating topics traditionally thought of as more advanced, such as coupling and the KMT strong approximation, option pricing via the equivalent martingale measure, and the isoperimetric inequality for Gaussian processes. The book is not ju

Generalising classical concepts of probability theory, the investigation of operator (semi)-stable laws as possible limit distributions of operator-normalized sums of i.i.d. random variable on finite-

First-passage properties underlie a wide range of stochastic processes, such as diffusion-limited growth, neuron firing and the triggering of stock options. This book provides a unified presentation of first-passage processes, which highlights its interrelations with electrostatics and the resulting powerful consequences. The author begins with a presentation of fundamental theory including the connection between the occupation and first-passage probabilities of a random walk, and the connection to electrostatics and current flows in resistor networks. The consequences of this theory are then developed for simple, illustrative geometries including the finite and semi-infinite intervals, fractal networks, spherical geometries and the wedge. Various applications are presented including neuron dynamics, self-organized criticality, diffusion-limited aggregation, the dynamics of spin systems and the kinetics of diffusion-controlled reactions. First-passage processes provide an appealing way

This graduate-level text and reference in probability, with numerous applications to several fields of science, presents nonmeasure-theoretic introduction to theory of Markov processes. The work also

The two parts of this book treat probability and statistics as mathematical disciplines and with the same degree of rigour as is adopted for other branches of applied mathematics at the level of a British honours degree. They contain the minimum information about these subjects that any honours graduate in mathematics ought to know. They are written primarily for general mathematicians, rather than for statistical specialists or for natural scientists who need to use statistics in their work. No previous knowledge of probability or statistics is assumed, though familiarity with calculus and linear algebra is required. The first volume takes the theory of probability sufficiently far to be able to discuss the simpler random processes, for example, queueing theory and random walks. The second volume deals with statistics, the theory of making valid inferences from experimental data, and includes an account of the methods of least squares and maximum likelihood; it uses the results of the

Major changes in this edition include the substitution of probabilistic arguments for combinatorial artifices, and the addition of new sections on branching processes, Markov chains, and the De Moivre

Introduction: The Nature of Probability Theory. The Sample Space. Elements of Combinatorial Analysis. Fluctuations in Coin Tossing and Random Walks. Combination of Events. Conditional Probability. Sto

Mathematical models of bond markets are of interest to researchers working in applied mathematics, especially in mathematical finance. This book concerns bond market models in which random elements are represented by Lévy processes. These are more flexible than classical models and are well suited to describing prices quoted in a discontinuous fashion. The book's key aims are to characterize bond markets that are free of arbitrage and to analyze their completeness. Nonlinear stochastic partial differential equations (SPDEs) are an important tool in the analysis. The authors begin with a relatively elementary analysis in discrete time, suitable for readers who are not familiar with finance or continuous time stochastic analysis. The book should be of interest to mathematicians, in particular to probabilists, who wish to learn the theory of the bond market and to be exposed to attractive open mathematical problems.

"The 4th edition of Ghahramani's book is replete with intriguing historical notes, insightful comments, and well-selected examples/exercises that, together, capture much of the essence of probability.

Branching processes are stochastic processes which represent the reproduction of particles, such as individuals within a population, and thereby model demographic stochasticity. In branching processes

This monograph aims to promote original mathematical methods to determine the invariant measure of two-dimensional random walks in domains with boundaries. Such processes arise in numerous application

Stochastic systems provide powerful abstract models for a variety of important real-life applications: for example, power supply, traffic flow, data transmission. They (and the real systems they model) are often subject to phase transitions, behaving in one way when a parameter is below a certain critical value, then switching behaviour as soon as that critical value is reached. In a real system, we do not necessarily have control over all the parameter values, so it is important to know how to find critical points and to understand system behaviour near these points. This book is a modern presentation of the 'semimartingale' or 'Lyapunov function' method applied to near-critical stochastic systems, exemplified by non-homogeneous random walks. Applications treat near-critical stochastic systems and range across modern probability theory from stochastic billiards models to interacting particle systems. Spatially non-homogeneous random walks are explored in depth, as they provide prototy

Branching Brownian motion (BBM) is a classical object in probability theory with deep connections to partial differential equations. This book highlights the connection to classical extreme value theory and to the theory of mean-field spin glasses in statistical mechanics. Starting with a concise review of classical extreme value statistics and a basic introduction to mean-field spin glasses, the author then focuses on branching Brownian motion. Here, the classical results of Bramson on the asymptotics of solutions of the F-KPP equation are reviewed in detail and applied to the recent construction of the extremal process of BBM. The extension of these results to branching Brownian motion with variable speed are then explained. As a self-contained exposition that is accessible to graduate students with some background in probability theory, this book makes a good introduction for anyone interested in accessing this exciting field of mathematics.

This book project concerns the connection between two branches of modern (theoretical and applied) probability - namely stochastic geometry and the Malliavin calculus of variations. Due to the strong

A simplified approach to Malliavin calculus adapted to Poisson random measures is developed and applied in this book. Called the “lent particle method” it is based on perturbation of the position of p

Will there be war with China? This book provides the most complete and accurate assessment of the probability of conflict between the United States and the rising Asian superpower. Equally important,

Revised and updated edition of the classic of advanced statistics. Uses concepts of gambling to develop important ideas in probability theory. "Strongly recommended." — Journal of the American Statist

Levy processes are the natural continuous-time analogue of random walks and form a rich class of stochastic processes around which a robust mathematical theory exists. Their application appears in the

One of the most notable features of nanometer scale CMOS technology is the increasing magnitude of variability of the key device parameters affecting performance of integrated circuits. The growth of variability can be attributed to multiple factors, including the difficulty of manufacturing control, the emergence of new systematic variation-generating mechanisms, and most importantly, the increase in atomic-scale randomness, where device operation must be described as a stochastic process. In addition to wide-sense stationary stochastic device variability and temperature variation, existence of non-stationary stochastic electrical noise associated with fundamental processes in integrated-circuit devices represents an elementary limit on the performance of electronic circuits.In an attempt to address these issues, Stochastic Process Variation in Deep-Submicron CMOS: Circuits and Algorithms offers unique combination of mathematical treatment of random process variation, electrical noise

Assuming only basic knowledge of probability theory and functional analysis, this book provides a self-contained introduction to Malliavin calculus and infinite-dimensional Brownian motion. In an effort to demystify a subject thought to be difficult, it exploits the framework of nonstandard analysis, which allows infinite-dimensional problems to be treated as finite-dimensional. The result is an intuitive, indeed enjoyable, development of both Malliavin calculus and nonstandard analysis. The main aspects of stochastic analysis and Malliavin calculus are incorporated into this simplifying framework. Topics covered include Brownian motion, Ornstein–Uhlenbeck processes both with values in abstract Wiener spaces, Lévy processes, multiple stochastic integrals, chaos decomposition, Malliavin derivative, Clark–Ocone formula, Skorohod integral processes and Girsanov transformations. The careful exposition, which is neither too abstract nor too theoretical, makes this book accessible to graduat

Does God exist?This is probably the most debated question in the history of mankind. Scholars, scientists, and philosophers have spent their lifetimes trying to prove or disprove the existence of God,

Rigorous exposition suitable for elementary instruction. Covers measure theory, axiomatization of probability theory, processes with independent increments, Markov processes and limit theorems for ran

This text introduces engineering students to probability theory and stochastic processes. Along with thorough mathematical development of the subject, the book presents intuitive explanations of key p

Control and communications engineers, physicists, and probability theorists, among others, will find this book unique. It contains a detailed development of approximation and limit theorems and method

This book provides an accessible introduction to stochastic processes in physics and describes the basic mathematical tools of the trade: probability, random walks, and Wiener and Ornstein-Uhlenbeck p

This book offers a superb overview of limit theorems and probability inequalities for sums of independent random variables. Unique in its combination of both classic and recent results, the book detai

How dangerous is our diet? How much in sports comes down to luck? When authorities pronounce that a given event is "highly likely"--well, how likely is that, really? Our lives are riddled with risk and uncertainty, and renowned statistician David Spiegelhalter has spent his career crunching data to decode the seemingly random events that define our experiences, as well as the numerous decisions we each make in the face of imperfect knowledge. In The Art of Uncertainty, he illuminates how all of us can do this better. Revealing the principles of probability--a field that informs everything from annuities to pandemics and climate change--he shows how we can measure our confidence, better evaluate cause and effect, and update our beliefs about the future in the face of constantly changing experience. Drawing on fascinating real-world examples, this is an essential guide to navigating uncertainty in a world that makes it inevitable.

Eleven types of armors were laboratory tested and their performances concerning the random placements of armors for two different surface slopes of rubble mound breakwater were compared. The three major categories of performance comparison are wave reflection, wave run-up and run-down, and armor stability. Due to the vast differences both in the setups of experiments and in the methodologies of data acquisition and analyses, the multitude of existing literatures yielded extremely scattering results, which are generally quite inconclusive and inconsistent, as well as incomparable. To cope with these problems the present research carries out the following: (1) Factors that affect armor behaviors are detailed and those that are modeled or not are indicated; (2) A real time laboratory system for data acquisition, monitoring, calibration, and analyses is developed, and the integrated program aims to provide accurate, convenient, and automated on-site processes; (3) Based upon the intrinsic

Heavy tails –extreme events or values more common than expected –emerge everywhere: the economy, natural events, and social and information networks are just a few examples. Yet after decades of progress, they are still treated as mysterious, surprising, and even controversial, primarily because the necessary mathematical models and statistical methods are not widely known. This book, for the first time, provides a rigorous introduction to heavy-tailed distributions accessible to anyone who knows elementary probability. It tackles and tames the zoo of terminology for models and properties, demystifying topics such as the generalized central limit theorem and regular variation. It tracks the natural emergence of heavy-tailed distributions from a wide variety of general processes, building intuition. And it reveals the controversy surrounding heavy tails to be the result of flawed statistics, then equips readers to identify and estimate with confidence. Over 100 exercises complete this e

Introduction to Reliability Engineering A complete revision of the classic text on reliability engineering, written by an expanded author team with increased industry perspective Introduction to Reliability Engineering provides a thorough and well-balanced overview of the fundamental aspects of reliability engineering and describes the role of probability and statistical analysis in predicting and evaluating reliability in a range of engineering applications. Covering both foundational theory and real-world practice, this classic textbook helps students of any engineering discipline understand key probability concepts, random variables and their use in reliability, Weibull analysis, system safety analysis, reliability and environmental stress testing, redundancy, failure interactions, and more. Extensively revised to meet the needs of today’s students, the Third Edition fully reflects current industrial practices and provides a wealth of new examples and problems that no