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This volume presents some of the most influential papers published by Rabi N. Bhattacharya, along with commentaries from international experts, demonstrating his knowledge, insight, and influence in the field of probability and its applications. For more than three decades, Bhattacharya has made significant contributions in areas ranging from theoretical statistics via analytical probability theory, Markov processes, and random dynamics to applied topics in statistics, economics, and geophysics. Selected reprints of Bhattacharya's papers are divided into three sections: Modes of Approximation, Large Times for Markov Processes, and Stochastic Foundations in Applied Sciences. The accompanying articles by the contributing authors not only help to position his work in the context of other achievements, but also provide a unique assessment of the state of their individual fields, both historically and for the next generation of researchers.Rabi N. Bhattacharya: Selected Papers will be a valuable resource for young researchers entering the diverse areas of study to which Bhattacharya has contributed. Established researchers will also appreciate this work as an account of both past and present developments and challenges for the future.

This volume contains the contributions of the participants of the 14th ISAAC congress, held at the University of São Paulo, Campus Ribeirão Preto, Brazil, on July 17-21, 2023. The papers, written by respected international experts, address recent results in mathematics, with a special focus on analysis. The volume constitutes a valuable resource on current research in mathematical analysis and its various interdisciplinary applications, both for specialists and non-specialists alike.

This collection honours Ron Doney's work and includes invited articles by his collaborators and friends. After an introduction reviewing Ron Doney's mathematical achievements and how they have influenced the field, the contributed papers cover both discrete-time processes, including random walks and variants thereof, and continuous-time processes, including Lévy processes and diffusions. A good number of the articles are focused on classical fluctuation theory and its ramifications, the area for which Ron Doney is best known.

This volume contains papers which were presented at the XV Latin American Congress of Probability and Mathematical Statistics (CLAPEM) in December 2019 in Mérida-Yucatán, México. They represent well the wide set of topics on probability and statistics that was covered at this congress, and their high quality and variety illustrates the rich academic program of the conference.

This is a volume in memory of Vladas Sidoravicius who passed away in 2019. Vladas has edited two volumes appeared in this series ('In and Out of Equilibrium') and is now honored by friends and colleagues with research papers reflecting Vladas'interests and contributions to probability theory.

This volume collects selected papers from the 8th High Dimensional Probability meeting held at Casa Matemática Oaxaca (CMO), Mexico. High Dimensional Probability (HDP) is an area of mathematics that includes the study of probability distributions and limit theorems in infinite-dimensional spaces such as Hilbert spaces and Banach spaces. The most remarkable feature of this area is that it has resulted in the creation of powerful new tools and perspectives, whose range of application has led to interactions with other subfields of mathematics, statistics, and computer science. These include random matrices, nonparametric statistics, empirical processes, statistical learning theory, concentration of measure phenomena, strong and weak approximations, functional estimation, combinatorial optimization, random graphs, information theory and convex geometry. The contributions in this volume show that HDP theory continues to thrive and develop new tools, methods, techniques and perspectives to analyze random phenomena.

This book contains the proceedings of the conference'Fractals in Graz 2001 - Analysis, Dynamics, Geometry, Stochastics'that was held in the second week of June 2001 at Graz University of Technology, in the capital of Styria, southeastern province of Austria. The scientific committee of the meeting consisted of M. Barlow (Vancouver), R. Strichartz (Ithaca), P. Grabner and W. Woess (both Graz), the latter two being the local organizers and editors of this volume. We made an effort to unite in the conference as well as in the present pro­ ceedings a multitude of different directions of active current work, and to bring together researchers from various countries as well as research fields that all are linked in some way with the modern theory of fractal structures. Although (or because) in Graz there is only a very small group working on fractal structures, consisting of'non-insiders', we hope to have been successful with this program of wide horizons. All papers were written upon explicit invitation by the editors, and we are happy to be able to present this representative panorama of recent work on poten­ tial theory, random walks, spectral theory, fractal groups, dynamic systems, fractal geometry, and more. The papers presented here underwent a refereeing process.

This volume consists of about half of the papers presented during a three-day seminar on stochastic processes held at Northwestern University in March 1982. This was the second of such yearly seminars aimed at bringing together a small group of researchers to discuss their current work in an informal atmosphere. The invited participants in this year's seminar were B. ATKINSON, R. BASS, K. BICHTELER, D. BURKHOLDER, K.L. CHUNG, J.L. DOOB, C. DOLEANS-DADE, H. FOLLMER, R.K. GETOOR, J. GLOVER, J. MITRO, D. MONRAD, E. PERKINS, J. PITMAN, Z. POP-STOJANOVIC, M.J. SHARPE, and J. WALSH. We thank them and the other participants for the lively atmosphere of the seminar. As mentioned above, the present volume is only a fragment of the work discussed at the seminar, the other work having been committed to other publications. The seminar was made possible through the enlightened support of the Air Force Office of Scientific Research, Grant No. 80-0252A. We are grateful to them as well as the publisher, Birkhauser, Boston, for their support and encouragement. E.C., Evanston, 1983 Seminar on stochastic Processes, 1982 Birkhauser, Boston, 1983 GERM FIELDS AND A CONVERSE TO THE STRONG MARKOV PROPERTY by BRUCE W. ATKINSON 1. Introduction The purpose of this paper is to give an intrinsic characterization of optional (i.e., stopping) times for the general germ Markov process, which includes the general right process as a special case. We proceed from the general to the specific.

This volume consists of twenty-four papers selected by the editors from the sixty-one papers presented at the 1st International Conference on Mathemati­ cal Methods in Reliability held at the Politehnica University of Bucharest from 16 to 19 September 1997. The papers have been divided into three sections: statistical methods, probabilistic methods, and special techniques and appli­ cations. Of course, as with any classification, some papers could be as well assigned to other sections. Problems in reliability are encountered in items in everyday usage. Relia­ bility is an important feature of household appliances, cars, telephones, power supplies, and so on, whether viewed from the vantage of the producer or the consumer. Important decisions are based on the reliability of the product. Obtaining systems that perform adequately for a specified period of time in a given environment is an important goal for both government and industry. Hence study and use of reliability theory, which can be applied in the research, development, and production phases of a system to enable the user to evaluate and improve performance, is a worthwhile venture. If reliability theory is to be useful, it must be quantitative in nature, because reliability must be demonstra­ ble. Subsequently probability and statistics, among others, play an important part in its development.

This volume contains a selection of invited papers, presented to the fourth In­ Statistical Analysis Based on the L1-Norm and Related ternational Conference on Methods, held in Neuchatel, Switzerland, from August 4-9, 2002. Organized jointly by the University of Illinois at Chicago (Gib Bassett), the Rutgers University (Regina Liu and Yehuda Vardi) and the University of Neuchatel (Yadolah Dodge), the conference brought together experts whose research deals with theory and ap­ plications involving the L1-Norm. The conference included invited and contributed talks as well as a tutorial on Quantile Regression. This volume includes 36 refereed invited papers under seven headings. Part one deals with Quantiles in all their forms and shapes. It includes papers on quantile functions in non-parametric multivariate analysis, and empirical applications of quantile regression. Much of the development in this direction follows from the fundamental paper by Koenker and Bassett in 1978. Financial and Time Series A nalysis follows the section on quantiles. Part three concerns Estimation, Testing and Characterization. Part four, Deep in the Data, deals with issues related to data depth. Part five addresses Classification questions. The problem of Density Estimation and Image Processing is discussed in Part six, and finally Part seven presents two environmental applications. The contributions represent clear evidence of important research involving theo­ retical issues and applications associated with the L1-Norm. It is my hope that the articles contained in this volume and its predecessors, published in 1987, 1992, and 1997, will stimulate interest among researchers.

A collection of 20 refereed research or review papers presented at a six-day seminar in Switzerland. The contributions focus on stochastic analysis, its applications to the engineering sciences, and stochastic methods in financial models, which was the subject of a minisymposium.

This volume consists of about half of the papers presented during a three-day seminar on stochastic processes held at Northwestern U- versity, Evanston. The seminar was the fourth of such yearly seminars aimed at bringing together a small group of researchers to discuss their current work in an informal atmosphere. The invited participants in the seminar were B.W. ATKINSON, R.M. BLUMENTHAL, K. BURDZY, D. BURKHOLDER, M. CRANSTON, C. DOLEANS''DADE, J.L. DOOB, N. FALKNER, P. FITZSIMMONS, J. GLOVER, F. KNIGHT, T. McCONNELL, J.B. MITRO, S. OREY, J. PITMAN, A.O. PITTENGER, Z. POP- STOJANOVIC, P. PROTTER, T. SALISBURY, M. SHARPE, C.T. SHIH, A. SZNITMAN, S.J. TAYLOR, J. WALSH, and R. WILLIAMS. We thank them and the other partiCipants for the lively seminar they created. The seminar was made possible through the partial support of the Air Force Office of Scientific Research via their Grant No. 82-0109 to Northwestern University. E.

The title High Dimensional Probability is an attempt to describe the many trib­ utaries of research on Gaussian processes and probability in Banach spaces that started in the early 1970's. In each of these fields it is necessary to consider large classes of stochastic processes under minimal conditions. There are rewards in re­ search of this sort. One can often gain deep insights, even about familiar processes, by stripping away details that in hindsight turn out to be extraneous. Many of the problems that motivated researchers in the 1970's were solved. But the powerful new tools created for their solution, such as randomization, isoperimetry, concentration of measure, moment and exponential inequalities, chaining, series representations and decoupling turned out to be applicable to other important areas of probability. They led to significant advances in the study of empirical processes and other topics in theoretical statistics and to a new ap­ proach to the study of aspects of Levy processes and Markov processes in general. Papers on these topics as well as on the continuing study of Gaussian processes and probability in Banach are included in this volume.

This volume contains 20 refereed research or review papers presented at the five-day Third Seminar on Stochastic Analysis, Random Fields and Applications which took place at the Centro Stefano Franscini (Monte Verità) in Ascona, Switzerland, from September 20 to 24, 1999. The seminar focused on three topics: fundamental aspects of stochastic analysis, physical modeling, and applications to financial engineering. The third topic was the subject of a mini-symposium on stochastic methods in financial models.

This volume contains the Proceedings of the Fourth Seminar on Stochastic Analy­ sis, Random Fields and Applications, which took place at the Centro Stefano Fran­ scini (Monte Verita) in Ascona (Ticino), Switzerland, from May 20 to 24, 2002. The first three editions of this conference occured in 1993, 1996 and 1999. The Seminar covered several topics: fundamental aspects of stochastic analysis, such as stochastic partial differential equations and random fields, and applications to current active fields such as probabilistic methods in fluid dynamics, biomathe­ matics, and financial modeling. As in the previous editions, this last topic was the subject of the Fourth Minisymposium on Stochastic Methods in Financial Models. These proceedings aim to present key aspects of these topics to a larger audience. All papers in this volume have been refereed. A major topic within Stochastic Analysis is the area of random fields which includes as particular cases, Gaussian random fields, stochastic partial differential equations (s. p. d. e.'s) and stochastic differential equations with values in Banach spaces. In this framework, interesting new developments were presented in the theory of Gaussian random fields on manifolds with applications to astrophysics and neurosciences. Moreover, with the aim of modeling certain very irregular phe­ nomena, a theory of s. p. d. e.'s driven by noises concentrated on hyperplanes was presented.

Statistics is strongly tied to applications in different scientific disciplines, and the most challenging statistical problems arise from problems in the sciences. In fact, the most innovative statistical research flows from the needs of applications in diverse settings. This volume is a testimony to the crucial role that statistics plays in scientific disciplines such as genetics and environmental sciences, among others. The articles in this volume range from human and agricultural genetic DNA research to carcinogens and chemical concentrations in the environment and to space debris and atmospheric chemistry. Also included are some articles on statistical methods which are sufficiently general and flexible to be applied to many practical situations. The papers were refereed by a panel of experts and the editors of the volume. The contributions are based on the talks presented at the Workshop on Statistics and the Sciences, held at the Centro Stefano Franscini in Ascona, Switzerland, during the week of May 23 to 28, 1999. The meeting was jointly organized by the Swiss Federal Institutes of Technology in Lausanne and Zurich, with the financial support of the Minerva Research Foundation. As the presentations at the workshop helped the participants recognize the po­ tential role that statistics can play in the sciences, we hope that this volume will help the reader to focus on the central role of statistics in the specific areas presented here and to extrapolate the results to further applications.

This volume consists of about half of the papers presented during a three-day seminar on stochastic processes. The seminar was the third of such yearly seminars aimed at bringing together a small group of researchers to discuss their current work in an informal atmosphere. The previous two seminars were held at Northwesterr. University, Evanston. This one was held at the University of Florida, Gainesville. The invited participants in the seminar were B. ATKINSON, K.L. CHUNG, C. DELLACHERIE, J.L. DOOB, E.B. DYNKIN, N. FALKNER, R.K. GETOOR, J. GLOVER, T. JEULIN, H. KASPI, T. McCONNELL, J. MITRO, E. PERKINS, Z. POP-STOJANOVIC, M. RAO, L.C.G. ROGERS, P. SALMINEN, M.J. SHARPE, S.R.S. VARADHAN, and J. WALSH. We thank them and the other participants for the lively atmosphere they have created. The seminar was made possible through the generous supports of the University of Florida, Department of Mathematics, and the Air Force Office of Scientific Research, Grant No. 82-0189, to Northwestern University. We are grateful for their support. Finally, we thank Professors Zoran POP-STOJANOVIC and Murali RAO for their time, effort, and kind hospitality in the organization of the seminar and during our stay in Gainesville.

This textbook provides an accessible approach to concepts and applications of stochastic processes ideal for a wide range of readers. This revised third edition features an intuitive reorganization with concrete topics introduced early on which are then used to demonstrate more abstract concepts in later chapters. The author has kept chapters short and independent from each other, with several of the longer chapters from previous editions now divided into smaller, more manageable parts. These changes build upon previous editions to allow readers even greater flexibility. The applications that are covered feature active areas of research within biological modeling, such as cancerous mutations, influenza evolution, drug resistance, and immune response. Important problems in fields such as engineering and mathematical physics are presented as well. These topics elegantly apply various classical stochastic models and are motivated throughout with many worked out examples. This third edition of Classical and Spatial Stochastic Processes is suitable as a textbook for a first course in stochastic processes at the upper-undergraduate or graduate level. Because of its accessible approach, it may also be used as a self-study resource for researchers and practitioners in mathematics, engineering, physics, and mathematical biology.

This text presents a collection of mathematical exercises with the aim of guiding readers to study topics in statistical physics, equilibrium thermodynamics, information theory, and their various connections. It explores essential tools from linear algebra, elementary functional analysis, and probability theory in detail and demonstrates their applications in topics such as entropy, machine learning, error-correcting codes, and quantum channels. The theory of communication and signal theory are also in the background, and many exercises have been chosen from the theory of wavelets and machine learning. Exercises are selected from a number of different domains, both theoretical and more applied. Notes and other remarks provide motivation for the exercises, and hints and full solutions are given for many. For senior undergraduate and beginning graduate students majoring in mathematics, physics, or engineering, this text will serve as a valuable guide as theymove on to more advanced work.

This textbook offers an approachable introduction to measure-theoretic probability, illustrating core concepts with examples from statistics and engineering. The author presents complex concepts in a succinct manner, making otherwise intimidating material approachable to undergraduates who are not necessarily studying mathematics as their major. Throughout, readers will learn how probability serves as the language in a variety of exciting fields. Specific applications covered include the coupon collector's problem, Monte Carlo integration in finance, data compression in information theory, and more. Measure-Theoretic Probability is ideal for a one-semester course and will best suit undergraduates studying statistics, data science, financial engineering, and economics who want to understand and apply more advanced ideas from probability to their disciplines. As a concise and rigorous introduction to measure-theoretic probability, it is also suitable for self-study.Prerequisites include a basic knowledge of probability and elementary concepts from real analysis.

This book extends the theory and applications of random evolutions to semi-Markov random media in discrete time, essentially focusing on semi-Markov chains as switching or driving processes. After giving the definitions of discrete-time semi-Markov chains and random evolutions, it presents the asymptotic theory in a functional setting, including weak convergence results in the series scheme, and their extensions in some additional directions, including reduced random media, controlled processes, and optimal stopping. Finally, applications of discrete-time semi-Markov random evolutions in epidemiology and financial mathematics are discussed. This book will be of interest to researchers and graduate students in applied mathematics and statistics, and other disciplines, including engineering, epidemiology, finance and economics, who are concerned with stochastic models of systems.

This book presents the latest developments in the theory of gradient flows in random walk spaces. A broad framework is established for a wide variety of partial differential equations on nonlocal models and weighted graphs. Within this framework, specific gradient flows that are studied include the heat flow, the total variational flow, and evolution problems of Leray-Lions type with different types of boundary conditions. With many timely applications, this book will serve as an invaluable addition to the literature in this active area of research.Variational and Diffusion Problems in Random Walk Spaces will be of interest to researchers at the interface between analysis, geometry, and probability, as well as to graduate students interested in exploring these areas.

Over the course of his distinguished career, Robert Strichartz (1943-2021) had a substantial impact on the field of analysis with his deep, original results in classical harmonic, functional, and spectral analysis, and in the newly developed analysis on fractals. This is the first volume of a tribute to his work and legacy, featuring chapters that reflect his mathematical interests, written by his colleagues and friends. An introductory chapter summarizes his broad and varied mathematical work and highlights his profound contributions as a mathematical mentor. The remaining articles are grouped into three sections – functional and harmonic analysis on Euclidean spaces, analysis on manifolds, and analysis on fractals – and explore Strichartz'contributions to these areas, as well as some of the latest developments.

This textbook, now in its third edition, offers a practical introduction to probability with statistical applications, covering material for both a first and second undergraduate probability course. The author focuses on essential concepts that every student should thoroughly understand. The content is organized into brief, easy-to-follow chapters, motivated by plenty of examples. The first part of the book focuses on classical discrete probability distributions, then goes on to study continuous distributions, confidence intervals, and statistical tests. The following section introduces more advanced concepts suitable for a second course in probability, such as random vectors and sums of random variables. The last part of the book is dedicated to mathematical statistics concepts such as estimation, sufficiency, Bayes'estimation, and multiple regression. This third edition includes a new chapter on combinatorics and a more distinct separation betweendiscrete and continuous distributions. Some of the longer chapters in the previous editions have been divided into shorter chapters to allow for more flexible teaching.Probability with Statistical Applications, Third Edition is intended for undergraduate students taking a first course in probability; later chapters are also suited for a second course in probability and mathematical statistics. Calculus is the only prerequisite; prior knowledge of probability is not required.

Probabilistic modeling and analysis of spatial telecommunication systems have never been more important than they are today. In particular, it is an essential research area for designing and developing next-generation communication networks that are based on multihop message transmission technology. These lecture notes provide valuable insights into the underlying mathematical discipline, stochastic geometry, introducing the theory, mathematical models and basic concepts. They also discuss the latest applications of the theory to telecommunication systems. The text covers several of the most fundamental aspects of quality of service: connectivity, coverage, interference, random environments, and propagation of malware. It especially highlights two important limiting scenarios of large spatial systems: the high-density limit and the ergodic limit. The book also features an analysis of extreme events and their probabilities based on the theory of large deviations.Lastly, it includes a large number of exercises offering ample opportunities for independent self-study.

Arbitrage Theory provides the foundation for the pricing of financial derivatives and has become indispensable in both financial theory and financial practice. This textbook offers a rigorous and comprehensive introduction to the mathematics of arbitrage pricing in a discrete-time, finite-state economy in which a finite number of securities are traded. In a first step, various versions of the Fundamental Theorem of Asset Pricing, i.e., characterizations of when a market does not admit arbitrage opportunities, are proved. The book then focuses on incomplete markets where the main concern is to obtain a precise description of the set of “market-consistent” prices for nontraded financial contracts, i.e. the set of prices at which such contracts could be transacted between rational agents.Both European-type and American-type contracts are considered. A distinguishing feature of this book is its emphasis on market-consistent prices and a systematic description of pricing rules, also at intermediate dates. The benefits of this approach are most evident in the treatment of American options, which is novel in terms of both the presentation and the scope, while also presenting new results.The focus on discrete-time, finite-state models makes it possible to cover all relevant topics while requiring only a moderate mathematical background on the part of the reader. The book will appeal to mathematical finance and financial economics students seeking an elementary but rigorous introduction to the subject; mathematics and physics students looking for an opportunity to get acquainted with a modern applied topic; and mathematicians, physicists and quantitatively inclined economists working or planning to work in the financial industry.

This volume features a collection of contributed articles and lecture notes from the XIII Symposium on Probability and Stochastic Processes, held at UNAM, Mexico, in December 2017.It is split into two main parts: the first one presents lecture notes of the course provided by Mauricio Duarte, followed by its second part which contains research contributions of some of the participants.

Continuing the authors'multivolume project, this text considers the theory of distributions from an applied perspective, demonstrating how effective a combination of analytic and probabilistic methods can be for solving problems in the physical and engineering sciences. Volume 1 covered foundational topics such as distributional and fractional calculus, the integral transform, and wavelets, and Volume 2 explored linear and nonlinear dynamics in continuous media. With this volume, the scope is extended to the use of distributional tools in the theory of generalized stochastic processes and fields, and in anomalous fractional random dynamics. Chapters cover topics such as probability distributions; generalized stochastic processes, Brownian motion, and the white noise; stochastic differential equations and generalized random fields; Burgers turbulence and passive tracer transport in Burgers flows; and linear, nonlinear, and multiscale anomalous fractional dynamics in continuous media. The needs of the applied-sciences audience are addressed by a careful and rich selection of examples arising in real-life industrial and scientific labs and a thorough discussion of their physical significance. Numerous illustrations generate a better understanding of the core concepts discussed in the text, and a large number of exercises at the end of each chapter expand on these concepts.Distributions in the Physical and Engineering Sciences is intended to fill a gap in the typical undergraduate engineering/physical sciences curricula, and as such it will be a valuable resource for researchers and graduate students working in these areas. The only prerequisites are a three-four semester calculus sequence (including ordinary differential equations, Fourier series, complex variables, and linear algebra), and some probability theory, but basic definitions and facts are covered as needed. An appendix also provides background material concerning the Dirac-delta and other distributions.

This book contains the notes of the lectures delivered at an Advanced Course on Combinatorial Matrix Theory held at Centre de Recerca Matemàtica (CRM) in Barcelona. These notes correspond to five series of lectures. The first series is dedicated to the study of several matrix classes defined combinatorially, and was delivered by Richard A. Brualdi. The second one, given by Pauline van den Driessche, is concerned with the study of spectral properties of matrices with a given sign pattern. Dragan Stevanović delivered the third one, devoted to describing the spectral radius of a graph as a tool to provide bounds of parameters related with properties of a graph. The fourth lecture was delivered by Stephen Kirkland and is dedicated to the applications of the Group Inverse of the Laplacian matrix. The last one, given by Ángeles Carmona, focuses on boundary value problems on finite networks with special in-depth on the M-matrix inverse problem.

Now in its second edition, this textbook serves as an introduction to probability and statistics for non-mathematics majors who do not need the exhaustive detail and mathematical depth provided in more comprehensive treatments of the subject. The presentation covers the mathematical laws of random phenomena, including discrete and continuous random variables, expectation and variance, and common probability distributions such as the binomial, Poisson, and normal distributions. More classical examples such as Montmort's problem, the ballot problem, and Bertrand's paradox are now included, along with applications such as the Maxwell-Boltzmann and Bose-Einstein distributions in physics.Key features in new edition:• 35 new exercises• Expanded section on the algebra of sets • Expanded chapters on probabilities to include more classical examples• New section on regression• Online instructors'manual containing solutions to all exercises

This volume contains lecture notes from the courses given by Vlad Bally and Rama Cont at the Barcelona Summer School on Stochastic Analysis (July 2012).The notes of the course by Vlad Bally, co-authored with Lucia Caramellino, develop integration by parts formulas in an abstract setting, extending Malliavin's work on abstract Wiener spaces. The results are applied to prove absolute continuity and regularity results of the density for a broad class of random processes.Rama Cont's notes provide an introduction to the Functional Itô Calculus, a non-anticipative functional calculus that extends the classical Itô calculus to path-dependent functionals of stochastic processes. This calculus leads to a new class of path-dependent partial differential equations, termed Functional Kolmogorov Equations, which arise in the study of martingales and forward-backward stochastic differential equations.This book will appeal to both young and senior researchers in probability and stochastic processes, as well as to practitioners in mathematical finance.

This textbook provides a coherent introduction to the main concepts and methods of one-parameter statistical inference. Intended for students of Mathematics taking their first course in Statistics, the focus is on Statistics for Mathematicians rather than on Mathematical Statistics. The goal is not to focus on the mathematical/theoretical aspects of the subject, but rather to provide an introduction to the subject tailored to the mindset and tastes of Mathematics students, who are sometimes turned off by the informal nature of Statistics courses. This book can be used as the basis for an elementary semester-long first course on Statistics with a firm sense of direction that does not sacrifice rigor. The deeper goal of the text is to attract the attention of promising Mathematics students.

This is a self-contained exposition by one of the leading experts in lattice theory, George Grätzer, presenting the major results of the last 70 years on congruence lattices of finite lattices, featuring the author's signature Proof-by-Picture method.Key features:• Insightful discussion of techniques to construct'nice'finite lattices with given congruence lattices and'nice'congruence-preserving extensions• Contains complete proofs, an extensive bibliography and index, and over 140 illustrations• This new edition includes two new parts on Planar Semimodular Lattices and The Order of Principle Congruences, covering the research of the last 10 yearsThe book is appropriate for a one-semester graduate course in lattice theory, and it is a practical reference for researchers studying lattices.Reviews of the first edition:'There exist a lot of interesting results in this area of lattice theory, and some of them are presented inthis book. [This] monograph…is an exceptional work in lattice theory, like all the contributions by this author. … The way this book is written makes it extremely interesting for the specialists in the field but also for the students in lattice theory. Moreover, the author provides a series of companion lectures which help the reader to approach the Proof-by-Picture sections.'(Cosmin Pelea, Studia Universitatis Babes-Bolyai Mathematica, Vol. LII (1), 2007)'The book is self-contained, with many detailed proofs presented that can be followed step-by-step. [I]n addition to giving the full formal details of the proofs, the author chooses a somehow more pedagogical way that he calls Proof-by-Picture, somehow related to the combinatorial (as opposed to algebraic) nature of many of the presented results. I believe that this book is a much-needed tool for any mathematician wishing a gentle introduction to the field of congruences representations offinite lattices, with emphasis on the more'geometric'aspects.'—Mathematical Reviews

This volume presents the lecture notes from two courses given by Davar Khoshnevisan and René Schilling, respectively, at the second Barcelona Summer School on Stochastic Analysis.René Schilling's notes are an expanded version of his course on Lévy and Lévy-type processes, the purpose of which is two-fold: on the one hand, the course presents in detail selected properties of the Lévy processes, mainly as Markov processes, and their different constructions, eventually leading to the celebrated Lévy-Itô decomposition. On the other, it identifies the infinitesimal generator of the Lévy process as a pseudo-differential operator whose symbol is the characteristic exponent of the process, making it possible to study the properties of Feller processes as space inhomogeneous processes that locally behave like Lévy processes. The presentation is self-contained, and includes dedicated chapters that review Markov processes, operator semigroups, random measures, etc. Inturn, Davar Khoshnevisan's course investigates selected problems in the field of stochastic partial differential equations of parabolic type. More precisely, the main objective is to establish an Invariance Principle for those equations in a rather general setting, and to deduce, as an application, comparison-type results. The framework in which these problems are addressed goes beyond the classical setting, in the sense that the driving noise is assumed to be a multiplicative space-time white noise on a group, and the underlying elliptic operator corresponds to a generator of a Lévy process on that group. This implies that stochastic integration with respect to the above noise, as well as the existence and uniqueness of a solution for the corresponding equation, become relevant in their own right. These aspects are also developed and supplemented by a wealth of illustrative examples.

Many probability books are written by mathematicians and have the built-in bias that the reader is assumed to be a mathematician coming to the material for its beauty. This textbook is geared towards beginning graduate students from a variety of disciplines whose primary focus is not necessarily mathematics for its own sake. Instead, A Probability Path is designed for those requiring a deep understanding of advanced probability for their research in statistics, applied probability, biology, operations research, mathematical finance and engineering. A one-semester course is laid out in an efficient and readable manner covering the core material. The first three chapters provide a functioning knowledge of measure theory. Chapter 4 discusses independence, with expectation and integration covered in Chapter 5, followed by topics on different modes of convergence, laws of large numbers with applications to statistics (quantile and distribution function estimation) and applied probability. Two subsequent chapters offer a careful treatment of convergence in distribution and the central limit theorem. The final chapter treats conditional expectation and martingales, closing with a discussion of two fundamental theorems of mathematical finance. Like Adventures in Stochastic Processes, Resnick's related and very successful textbook, A Probability Path is rich in appropriate examples, illustrations and problems and is suitable for classroom use or self-study. The present uncorrected, softcover reprint is designed to make this classic textbook available to a wider audience. This book is different from the classical textbooks on probability theory in that it treats the measure theoretic background not as a prerequisite but as an integral part of probability theory. The result is that the reader gets a thorough and well-structured framework needed to understand the deeper concepts of current day advanced probability as itis used in statistics, engineering, biology and finance.... The pace of the book is quick and disciplined. Yet there are ample examples sprinkled over the entire book and each chapter finishes with a wealthy section of inspiring problems. —Publications of the International Statistical Institute This textbook offers material for a one-semester course in probability, addressed to students whose primary focus is not necessarily mathematics.... Each chapter is completed by an exercises section. Carefully selected examples enlighten the reader in many situations. The book is an excellent introduction to probability and its applications. —Revue Roumaine de Mathématiques Pures et Appliquées

Branching processes form one of the classical fields of applied probability and are still an active area of research. The field has by now grown so large and diverse that a complete and unified treat­ ment is hardly possible anymore, let alone in one volume. So, our aim here has been to single out some of the more recent developments and to present them with sufficient background material to obtain a largely self-contained treatment intended to supplement previous mo­ nographs rather than to overlap them. The body of the text is divided into four parts, each of its own flavor. Part A is a short introduction, stressing examples and applications. In Part B we give a self-contained and up-to-date pre­ sentation of the classical limit theory of simple branching processes, viz. the Gal ton-Watson ( Bienayme-G-W) process and i ts continuous time analogue. Part C deals with the limit theory of Il!arkov branching processes with a general set of types under conditions tailored to (multigroup) branching diffusions on bounded domains, a setting which also covers the ordinary multitype case. Whereas the point of view in Parts A and B is quite pedagogical, the aim of Part C is to treat a large subfield to the highest degree of generality and completeness possi'ble. Thus the exposition there is at times quite technical.

ANOVA models involving random effects have found widespread application to experimental design in varied fields such as biology, econometrics, and engineering. Volume I of this two-part work is a comprehensive presentation of methods and techniques for point estimation, interval estimation, and hypotheses tests for linear models involving random effects. Volume I examines models with balanced data (orthogonal models); Volume II studies models with unbalanced data (non-orthogonal models). Accessible to readers with a modest mathematical and statistical background, the work will appeal to a broad audience of graduate students, researchers, and practitioners. It can be used as a graduate text or as a self-study reference.

A self-avoiding walk is a path on a lattice that does not visit the same site more than once. In spite of this simple definition, many of the most basic questions about this model are difficult to resolve in a mathematically rigorous fashion. In particular, we do not know much about how far an n­ step self-avoiding walk typically travels from its starting point, or even how many such walks there are. These and other important questions about the self-avoiding walk remain unsolved in the rigorous mathematical sense, although the physics and chemistry communities have reached consensus on the answers by a variety of nonrigorous methods, including computer simulations. But there has been progress among mathematicians as well, much of it in the last decade, and the primary goal of this book is to give an account of the current state of the art as far as rigorous results are concerned. A second goal of this book is to discuss some of the applications of the self-avoiding walk in physics and chemistry, and to describe some of the nonrigorous methods used in those fields. The model originated in chem­ istry several decades ago as a model for long-chain polymer molecules. Since then it has become an important model in statistical physics, as it exhibits critical behaviour analogous to that occurring in the Ising model and related systems such as percolation.

Overview This book is intended as a textbook in probability for graduate students in math­ ematics and related areas such as statistics, economics, physics, and operations research. Probability theory is a'difficult'but productive marriage of mathemat­ ical abstraction and everyday intuition, and we have attempted to exhibit this fact. Thus we may appear at times to be obsessively careful in our presentation of the material, but our experience has shown that many students find them­ selves quite handicapped because they have never properly come to grips with the subtleties of the definitions and mathematical structures that form the foun­ dation of the field. Also, students may find many of the examples and problems to be computationally challenging, but it is our belief that one of the fascinat­ ing aspects of prob ability theory is its ability to say something concrete about the world around us, and we have done our best to coax the student into doing explicit calculations, often in the context of apparently elementary models. The practical applications of probability theory to various scientific fields are far-reaching, and a specialized treatment would be required to do justice to the interrelations between prob ability and any one of these areas. However, to give the reader a taste of the possibilities, we have included some examples, particularly from the field of statistics, such as order statistics, Dirichlet distri­ butions, and minimum variance unbiased estimation.

Quite apart from the fact that percolation theory had its orlgln in an honest applied problem (see Hammersley and Welsh (1980)), it is a source of fascinating problems of the best kind a mathematician can wish for: problems which are easy to state with a minimum of preparation, but whose solutions are (apparently) difficult and require new methods. At the same time many of the problems are of interest to or proposed by statistical physicists and not dreamt up merely to demons~te ingenuity. Progress in the field has been slow. Relatively few results have been established rigorously, despite the rapidly growing literature with variations and extensions of the basic model, conjectures, plausibility arguments and results of simulations. It is my aim to treat here some basic results with rigorous proofs. This is in the first place a research monograph, but there are few prerequisites; one term of any standard graduate course in probability should be more than enough. Much of the material is quite recent or new, and many of the proofs are still clumsy. Especially the attempt to give proofs valid for as many graphs as possible led to more complications than expected. I hope that the Applications and Examples provide justifi­ cation for going to this level of generality.

Since the publication of the first edition of this seminar book in 1994, the theory and applications of extremes and rare events have enjoyed an enormous and still increasing interest. The intention of the book is to give a mathematically oriented development of the theory of rare events underlying various applications. This characteristic of the book was strengthened in the second edition by incorporating various new results on about 130 additional pages. Part II, which has been added in the second edition, discusses recent developments in multivariate extreme value theory. Particularly notable is a new spectral decomposition of multivariate distributions in univariate ones which makes multivariate questions more accessible in theory and practice. One of the most innovative and fruitful topics during the last decades was the introduction of generalized Pareto distributions in the univariate extreme value theory. Such a statistical modelling of extremes is now systematically developed in the multivariate framework.

This book is intended as a one-semester first course in probability and statistics, requiring only a knowledge of calculus. It will be useful for students majoring in a number of disciplines:for example,biology, computer science, electrical engineer­ ing, mathematics, and physics. Many good texts in probability and statistics are intended for a one-year course and consist of a large number of topics. In this book, the number of topics is dras­ tically reduced. We concentrate instead on several important concepts that every student should understand and be able to apply in an interesting and useful way. Thus statistics is introduced at an early stage. The presentation focuses on topics in probability and statistics and tries to min­ imize the difficulties students often have with calculus. Theory therefore is kept to a minimum and interesting examples are provided throughout. Chapter I contains the basic rules of probability and conditional probability with some interesting ap­ plications such asBayes'rule and the birthday problem. In Chapter 2 discrete and continuous random variables, expectation and variance are introduced. This chapter is mostly computational with a few probability concepts and many applications of calculus. In Chapters 3 and 4 we get to the heart of the subject: binomial distribu­ tion, normal approximation of the binomial, Poisson distribution, the Law of Large Numbers and the Central Limit Theorem. Wealso cover the Poisson approximation of the binomial (in a nonstandard way) and the Poisson scattering theorem.

The statistical analysis of extreme data is important for various disciplines, including hydrology, insurance, finance, engineering and environmental sciences. This book provides a self-contained introduction to the parametric modeling, exploratory analysis and statistical interference for extreme values. The entire text of this third edition has been thoroughly updated and rearranged to meet the new requirements. Additional sections and chapters, elaborated on more than 100 pages, are particularly concerned with topics like dependencies, the conditional analysis and the multivariate modeling of extreme data. Parts I–III about the basic extreme value methodology remain unchanged to some larger extent, yet notable are, e.g., the new sections about'An Overview of Reduced-Bias Estimation'(co-authored by M.I. Gomes),'The Spectral Decomposition Methodology', and'About Tail Independence'(co-authored by M. Frick), and the new chapter about'Extreme Value Statistics of Dependent Random Variables'(co-authored by H. Drees). Other new topics, e.g., a chapter about'Environmental Sciences', (co--authored by R.W. Katz), are collected within Parts IV–VI.

'…the text is user friendly to the topics it considers and should be very accessible…Instructors and students of statistical measure theoretic courses will appreciate the numerous informative exercises; helpful hints or solution outlines are given with many of the problems. All in all, the text should make a useful reference for professionals and students.'—The Journal of the American Statistical Association

A highly readable introduction to stochastic integration and stochastic differential equations, this book combines developments of the basic theory with applications. It is written in a style suitable for the text of a graduate course in stochastic calculus, following a course in probability.Using the modern approach, the stochastic integral is defined for predictable integrands and local martingales; then It's change of variable formula is developed for continuous martingales. Applications include a characterization of Brownian motion, Hermite polynomials of martingales, the Feynman–Kac functional and the Schrödinger equation. For Brownian motion, the topics of local time, reflected Brownian motion, and time change are discussed.New to the second edition are a discussion of the Cameron–Martin–Girsanov transformation and a final chapter which provides an introduction to stochastic differential equations, as well as many exercises for classroom use.This book willbe a valuable resource to all mathematicians, statisticians, economists, and engineers employing the modern tools of stochastic analysis.The text also proves that stochastic integration has made an important impact on mathematical progress over the last decades and that stochastic calculus has become one of the most powerful tools in modern probability theory. —Journal of the American Statistical Association An attractive text…written in [a] lean and precise style…eminently readable. Especially pleasant are the care and attention devoted to details… A very fine book.—Mathematical Reviews

The 1990 Seminar on Stochastic Processes was held at the University of British Columbia from May 10 through May 12, 1990. This was the tenth in a series of annual meetings which provide researchers with the opportunity to discuss current work on stochastic processes in an informal and enjoyable atmosphere. Previous seminars were held at Northwestern University, Princeton University, the Univer­ sity of Florida, the University of Virginia and the University of California, San Diego. Following the successful format of previous years, there were five invited lectures, delivered by M. Marcus, M. Vor, D. Nualart, M. Freidlin and L. C. G. Rogers, with the remainder of the time being devoted to informal communications and workshops on current work and problems. The enthusiasm and interest of the participants created a lively and stimulating atmosphere for the seminar. A sample of the research discussed there is contained in this volume. The 1990 Seminar was made possible by the support of the Natural Sciences and Engin~ring Research Council of Canada, the Southwest University Mathematics Society of British Columbia, and the University of British Columbia. To these entities and the organizers of this year's conference, Ed Perkins and John Walsh, we extend oul'thanks. Finally, we acknowledge the support and assistance of the staff at Birkhauser Boston.

The contents of this monograph approximate the lectures I gave In a graduate course at Stanford University in the first half of 1981. But the material has been thoroughly reorganized and rewritten. The purpose is to present a modern version of the theory of stochastic in­ tegration, comprising but going beyond the classical theory, yet stopping short of the latest discontinuous (and to some distracting) ramifications. Roundly speaking, integration with respect to a local martingale with continuous paths is the primary object of study here. We have decided to include some results requiring only right continuity of paths, in order to illustrate the general methodology. But it is possible for the reader to skip these extensions without feeling lost in a wilderness of generalities. Basic probability theory inclusive of martingales is reviewed in Chapter 1. A suitably prepared reader should begin with Chapter 2 and consult Chapter 1 only when needed. Occasionally theorems are stated without proof but the treatmcnt is aimed at self-containment modulo the in­ evitable prerequisites. With considerable regret I have decided to omit a discussion of stochastic differential equations. Instead, some other ap­ plications of the stochastic calculus are given; in particular Brownian local time is treated in dctail to fill an unapparent gap in the literature. x I PREFACE The applications to storage theory discussed in Section 8. 4 are based on lectures given by J. Michael Harrison in my class.

Intended as a self-contained introduction to measure theory, this textbook also includes a comprehensive treatment of integration on locally compact Hausdorff spaces, the analytic and Borel subsets of Polish spaces, and Haar measures on locally compact groups. Measure Theory provides a solid background for study in both harmonic analysis and probability theory and is an excellent resource for advanced undergraduate and graduate students in mathematics. The prerequisites for this book are courses in topology and analysis.

According to Leo Breiman (1968), probability theory has a right and a left hand. The right hand refers to rigorous mathematics, and the left hand refers to'proba­ bilistic thinking'. The combination of these two aspects makes probability theory one of the most exciting fields in mathematics. One can study probability as a purely mathematical enterprise, but even when you do that, all the concepts that arise do have a meaning on the intuitive level. For instance, we have to define what we mean exactly by independent events as a mathematical concept, but clearly, we all know that when we flip a coin twice, the event that the first gives heads is independent of the event that the second gives tails. Why have I written this book? I have been teaching probability for more than fifteen years now, and decided to do something with this experience. There are already many introductory texts about probability, and there had better be a good reason to write a new one. I will try to explain my reasons now.