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This volume collects selected papers from the Ninth High Dimensional Probability Conference, held virtually from June 15-19, 2020. These papers cover a wide range of topics and demonstrate how high-dimensional probability remains an active area of research with applications across many mathematical disciplines. Chapters are organized around four general topics: inequalities and convexity; limit theorems; stochastic processes; and high-dimensional statistics. High Dimensional Probability IX will be a valuable resource for researchers in this area.

This volume contains the proceedings of the CRM Workshops on Probabilistic Methods in Spectral Geometry and PDE, held from August 22–26, 2016 and Probabilistic Methods in Topology, held from November 14–18, 2016 at the Centre de Recherches Mathématiques, Université de Montréal, Montréal, Quebec, Canada. Probabilistic methods have played an increasingly important role in many areas of mathematics, from the study of random groups and random simplicial complexes in topology, to the theory of random Schrödinger operators in mathematical physics. The workshop on Probabilistic Methods in Spectral Geometry and PDE brought together some of the leading researchers in quantum chaos, semi-classical theory, ergodic theory and dynamical systems, partial differential equations, probability, random matrix theory, mathematical physics, conformal field theory, and random graph theory. Its emphasis was on the use of ideas and methods from probability in different areas, such as quantum chaos (study of spectra and eigenstates of chaotic systems at high energy); geometry of random metrics and related problems in quantum gravity; solutions of partial differential equations with random initial conditions. The workshop Probabilistic Methods in Topology brought together researchers working on random simplicial complexes and geometry of spaces of triangulations (with connections to manifold learning); topological statistics, and geometric probability; theory of random groups and their properties; random knots; and other problems. This volume covers recent developments in several active research areas at the interface of Probability, Semiclassical Analysis, Mathematical Physics, Theory of Automorphic Forms and Graph Theory.

This volume contains the proceedings of the XII Symposium of Probability and Stochastic Processes which took place at Universidad Autonoma de Yucatan in Merida, Mexico, on November 16–20, 2015. This meeting was the twelfth meeting in a series of ongoing biannual meetings aimed at showcasing the research of Mexican probabilists as well as promote new collaborations between the participants.The book features articles drawn from different research areas in probability and stochastic processes, such as: risk theory, limit theorems, stochastic partial differential equations, random trees, stochastic differential games, stochastic control, and coalescence. Two of the main manuscripts survey recent developments on stochastic control and scaling limits of Markov-branching trees, written by Kazutoshi Yamasaki and Bénédicte Haas, respectively. The research-oriented manuscripts provide new advances in active research fields in Mexico.The wide selection of topics makes the book accessible to advanced graduate students and researchers in probability and stochastic processes.

This book presents articles on original material from invited talks given at the “IMS Workshop on Applied Probability” organized by the Institute of Mathematical Sciences at the Chinese University of Hong Kong in May 1999. The goal of the workshop was to promote research in applied probability for local mathematicians and engineers and to foster exchange with experts from other parts of the world. The main themes were mathematical finance and stochastic networks. The topics range from the theoretical study, e.g., ergodic theory and diffusion processes, to very practical problems, such as convertible bonds with market risk and insider trading. The wide scope of coverage in the book make it a helpful reference for graduate students and researchers, and for practitioners working in mathematical finance.

GSP 286 contains 21 peer-reviewed papers on engineering uncertainties and reliability analysis in honor of Wilson H. Tang, Ph.D., P.E., S.E.

Designed for precollege teachers by a collaborative of teachers, educators, and mathematicians, Probability and Games is based on a course offered in the Summer School Teacher Program at the Park City Mathematics Institute. This course leads participants through an introduction to probability and statistics, with particular focus on conditional probability, hypothesis testing, and the mathematics of election analysis. These ideas are tied together through low-threshold entry points including work with real and fake coin-flipping data, short games that lead to key concepts, and inroads to connecting the topics to number theory and algebra. But this book isn't a “course” in the traditional sense. It consists of a carefully sequenced collection of problem sets designed to develop several interconnected mathematical themes. These materials provide participants with the opportunity for authentic mathematical discovery—participants build mathematical structures by investigating patterns, use reasoning to test and formalize their ideas, offer and negotiate mathematical definitions, and apply their theories and mathematical machinery to solve problems. Probability and Games is a volume of the book series “IAS/PCMI—The Teacher Program Series” published by the American Mathematical Society. Each volume in this series covers the content of one Summer School Teacher Program year and is independent of the rest.

This volume presents original research articles and extended surveys related to the mathematical interest and work of Jean-Michel Bismut. His outstanding contributions to probability theory and global analysis on manifolds have had a profound impact on several branches of mathematics in the areas of control theory, mathematical physics and arithmetic geometry. Contributions by:K. BehrendN. Bergeron S. K. Donaldson J. Dubédat B. DuplantierG. Faltings E. Getzler G. KingsR. MazzeoJ. MillsonC. MoeglinW. MüllerR. RhodesD. Rössler S. Sheffield A. Teleman G. Tian K-I. YoshikawaH. Weiss W. Werner The collection is a valuable resource for graduate students and researchers in these fields.

This volume is based on talks given at a conference celebrating Stanislav Molchanov's 65th birthday held in June of 2005 at the Centre de Recherches Mathématiques (Montreal, QC, Canada). The meeting brought together researchers working in an exceptionally wide range of topics reflecting the quality and breadth of Molchanov's past and present research accomplishments. This collection of survey and research papers gives a glance of the profound consequences of Molchanov's contributions in stochastic differential equations, spectral theory for deterministic and random operators, localization and intermittency, mathematical physics and optics, and other topics.

This volume is comprised of two parts: the first contains articles by S. N. Evans, F. Ledrappier, and Figà-Talomanaca. These articles arose from a Centre de Recherches de Mathématiques (CRM) seminar entitiled, “Topics in Probability on Lie Groups: Boundary Theory”. Evans gives a synthesis of his pre-1992 work on Gaussian measures on vector spaces over a local field. Ledrappier uses the freegroup on $d$ generators as a paradigm for results on the asymptotic properties of random walks and harmonic measures on the Martin boundary. These articles are followed by a case study by Figà-Talamanca using Gelfand pairs to study a diffusion on a compact ultrametric space. The second part of the book is an appendix to the book Compactifications of Symmetric Spaces (Birkhauser) by Y. Guivarc'h and J. C. Taylor. This appendix consists of an article by each author and presents the contents of this book in a more algebraic way. L. Ji and J.-P. Anker simplifies some of their results on the asymptotics of the Green function that were used to compute Martin boundaries. And Taylor gives a self-contained account of Martin boundary theory for manifolds using the theory of second order strictly elliptic partial differential operators.

This book brings together selected peer-reviewed contributions from various research fields in statistics, and highlights the diverse approaches and analyses related to real-life phenomena. Major topics covered in this volume include, but are not limited to, bayesian inference, likelihood approach, pseudo-likelihoods, regression, time series, and data analysis as well as applications in the life and social sciences.The software packages used in the papers are made available by the authors.This book is a result of the 47th Scientific Meeting of the Italian Statistical Society, held at the University of Cagliari, Italy, in 2014.

This volume collects selected papers from the 7th High Dimensional Probability meeting held at the Institut d'Études Scientifiques de Cargèse (IESC) in Corsica, France. 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, and random graphs.The contributions in this volume show that HDP theory continues to thrive and develop new tools, methods, techniques andperspectives to analyze random phenomena.

This book brings a reader to the cutting edge of several important directions of the contemporary probability theory, which in many cases are strongly motivated by problems in statistical physics. The authors of these articles are leading experts in the field and the reader will get an exceptional panorama of the field from the point of view of scientists who played, and continue to play, a pivotal role in the development of the new methods and ideas, interlinking it with geometry, complex analysis, conformal field theory, etc., making modern probability one of the most vibrant areas in mathematics.

This book presents the proceedings of the 14th International Probabilistic Workshop that was held in Ghent, Belgium in December 2016. Probabilistic methods are currently of crucial importance for research and developments in the field of engineering, which face challenges presented by new materials and technologies and rapidly changing societal needs and values. Contemporary needs related to, for example, performance-based design, service-life design, life-cycle analysis, product optimization, assessment of existing structures and structural robustness give rise to new developments as well as accurate and practically applicable probabilistic and statistical engineering methods to support these developments. These proceedings are a valuable resource for anyone interested in contemporary developments in the field of probabilistic engineering applications.

The series of advanced courses initiated in Séminaire de Probabilités XXXIII continues with a course by Ivan Nourdin on Gaussian approximations using Malliavin calculus. The Séminaire also occasionally publishes a series of contributions on a unifying subject; in this spirit, selected participants to the September 2011 Conference on Stochastic Filtrations, held in Strasbourg and organized by Michel Émery, have also contributed to the present volume. The rest of the work covers a wide range of topics, such as stochastic calculus and Markov processes, random matrices and free probability, and combinatorial optimization.

​Random Matrices are one of the major research areas in modern probability theory, due to their prominence in many different fields such as nuclear physics, statistics, telecommunication, free probability, non-commutative geometry, and dynamical systems. A great deal of recent work has focused on the study of spectra of large random matrices on the one hand and on iterated random functions, especially random difference equations, on the other. However, the methods applied in these two research areas are fairly dissimilar. Motivated by the idea that tools from one area could potentially also be helpful in the other, the volume editors have selected contributions that present results and methods from random matrix theory as well as from the theory of iterated random functions. This work resulted from a workshop that was held in Münster, Germany in 2011. The aim of the workshop was to bring together researchers from two fields of probability theory: random matrix theory and the theory of iterated random functions. Random matrices play fundamental, yet very different roles in the two fields. Accordingly, leading figures and young researchers gave talks on their field of interest that were also accessible to a broad audience.

Proceedings of the Ispra Course held at the Joint Research Centre, Ispra, Italy, November 16-19, 1987

The analysis of experimental data resulting from some underlying random process is a fundamental part of most scientific research. Probability Theory and Statistics have been developed as flexible tools for this analyis, and have been applied successfully in various fields such as Biology, Economics, Engineering, Medicine or Psychology. However, traditional techniques in Probability and Statistics were devised to model only a singe source of uncertainty, namely randomness. In many real-life problems randomness arises in conjunction with other sources, making the development of additional'softening'approaches essential. This book is a collection of papers presented at the 2nd International Conference on Soft Methods in Probability and Statistics (SMPS'2004) held in Oviedo, providing a comprehensive overview of the innovative new research taking place within this emerging field.

This symposium is the seventh of a series of IUTAM sponsored symposia which focus on probabilistic methods in mechanics. It is the sequel to the series of meetings in Coventry, UK (1972), Southhampton, UK (1976), Frankfurt/Oder, Germany (1982), Stockholm, Sweden (1984), Innsbruck/Igls, Austria (1987), and Turin, Italy (1991). The symposium focused on advances in the area of probabilistic mechanics with direct application to structural reliability issues. The contributed papers address collectively the four components of a structural reliability problem. They are: characterization of stochastic loads, description of material properties in terms of fatigue and fracture, response determination, and quantitative assessment of the reliability of the structural system. Four Keynote Lectures by V. Bolotin (Russia), o. Ditlevsen (Denmark), R. Heller (USA), and F. Ziegler (Austria) were delivered; the remaining contributed papers were organized in ten technical sessIons. A reception was hosted by Dr. Y. Wu the first day of the symposium; the second day of the symposium a banquet was hosted by Dr. P. Spanos, with Dr. N. Abramson serving as the banquet speaker. Closing remarks were provided by the IUTAM Secretary General, Dr. F. Ziegler.

The two volumes to which this is apreface consist of the Proceedings of the Second International Conference on History and Philosophy of Science. The Conference was organized by the Joint Commission of the International Union of History and Philosophy of Science (IUHPS) under the auspices of the IUHPS, the Italian Society for Logic and Philosophy of Science, and the Domus Galilaeana of Pisa, headed by Professor Vincenzo Cappelletti. Domus Galilaeana also served as the host institution, with some help from the University of Pisa. The Conference took place in Pisa, Italy, on September 4-8, 1978. The editors of these two volumes of the Proceedings of the Pisa Conference acknowledge with gratitude the help by the different sponsoring organizations, and in the first place that by both Divisions of the IUHPS, which made the Conference possible. A special recognition is due to Professor Evandro Agazzi, President of the Italian Society for Logic and Philosophy of Science, who was co­ opted as an additional member of the Organizing Committee. This committee was otherwise identical with the Joint Commission, whose members were initially John Murdoch, John North, Arpad Szab6, Robert Butts, Jaakko Hintikka, and Vadim Sadovsky. Later, Erwin Hiebert and Lubos Novy were appointed as additional members.

This volume describes the current state of knowledge of random spatial processes, particularly those arising in physics. The emphasis is on survey articles which describe areas of current interest to probabilists and physicists working on the probability theory of phase transition. Special attention is given to topics deserving further research. The principal contributions by leading researchers concern the mathematical theory of random walk, interacting particle systems, percolation, Ising and Potts models, spin glasses, cellular automata, quantum spin systems, and metastability. The level of presentation and review is particularly suitable for postgraduate and postdoctoral workers in mathematics and physics, and for advanced specialists in the probability theory of spatial disorder and phase transition.

This is a collection of papers by participants at High Dimensional Probability VI Meeting held from October 9-14, 2011 at the Banff International Research Station in Banff, Alberta, Canada. 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 areas of mathematics, statistics, and computer science. These include random matrix theory, nonparametric statistics, empirical process theory, statistical learning theory, concentration of measure phenomena, strong and weak approximations, distribution function estimation in high dimensions, combinatorial optimization, and random graph theory. The papers in this volume show that HDP theory continues to develop new tools, methods, techniques and perspectives to analyze the random phenomena. Both researchers and advanced students will find this book of great use for learning about new avenues of research.​

This volume contains the current research in quantum probability, infinite dimensional analysis and related topics. Contributions by experts in these fields highlight the latest developments and interdisciplinary connections with classical probability, stochastic analysis, white noise analysis, functional analysis and quantum information theory.This diversity shows how research in quantum probability and infinite dimensional analysis is very active and strongly involved in the modern mathematical developments and applications.Tools and techniques presented here will be of great value to researchers.

This book contains selected and refereed contributions to the'Inter­ national Symposium on Probability and Bayesian Statistics'which was orga­ nized to celebrate the 80th birthday of Professor Bruno de Finetti at his birthplace Innsbruck in Austria. Since Professor de Finetti died in 1985 the symposium was dedicated to the memory of Bruno de Finetti and took place at Igls near Innsbruck from 23 to 26 September 1986. Some of the pa­ pers are published especially by the relationship to Bruno de Finetti's scientific work. The evolution of stochastics shows growing importance of probability as coherent assessment of numerical values as degrees of believe in certain events. This is the basis for Bayesian inference in the sense of modern statistics. The contributions in this volume cover a broad spectrum ranging from foundations of probability across psychological aspects of formulating sub­ jective probability statements, abstract measure theoretical considerations, contributions to theoretical statistics and stochastic processes, to real applications in economics, reliability and hydrology. Also the question is raised if it is necessary to develop new techniques to model and analyze fuzzy observations in samples. The articles are arranged in alphabetical order according to the family name of the first author of each paper to avoid a hierarchical ordering of importance of the different topics. Readers interested in special topics can use the index at the end of the book as guide.

From August 21 through August 27, 1989 the Nato Advanced Research Workshop Probabilistic Methods in Quantum Field Theory and Quantum Gravity'was held at l'Institut d'Etudes Scientifiques, Cargese, France. This publication is the Proceedings of this workshop. The purpose of the workshop was to bring together a group of scientists who have been at the forefront of the development of probabilistic methods in Quantum Field Theory and Quantum Gravity. The original thought was to put emphasis on the introduction of stochastic processes in the understanding of Euclidean Quantum Field Theory, with also some discussion of recent progress in the field of stochastic numerical methods. During the final preparation of the meeting we broadened the scope to include all those Euclidean Quantum Field Theory descriptions that make direct reference to concepts from probability theory and statistical mechanics. Several of the main contributions centered around a more rigorous discussion of stochastic processes for the formulation of Euclidean Quantum Field Theory. These rather stringent mathematical approaches were contrasted with the more heuristic stochastic quantization scheme developed in 1981 by Parisi and Wu: Stochastic quan­ tization, its intrinsic BRST -structure and stochastic regularization appeared in many disguises and in connection with several different problems throughout the workshop.

Since 1972 the Institute of Mathematics and the Committee of Mathematics of the Polish Academy of Sciences organize annually con­ ferences on mathematical statistics in Wisla. The 1978 conference, supported also by the University of Wroclaw,was held in Wisla from December 7 to December 13 and attended by around 100 participants from 11 countries. K. Urbanik, Rector of the University of Wroclaw, was the honorary chairman of the conference. Traditionally at these conferences there are presented results on mathematical statistics and related fields obtained in Poland during the year of the conference as well as results presented by invited scholars from other countries. In 1978 invitations to present talks were accepted by 20 e~inent statisticians and probabilists. The topics of the invited lectures and contributed papers included theoretical statistics with a broad cover of the theory of linear models, inferences from stochastic processes, probability theory and applications to biology and medicine. In these notes there appear papers submitted by 30 participants of the conference. During the conference, on December 9, there was held a special session of the Polish Mathematical Society on the occasion of elect­ ing Professor Jerzy Neyman the honorary member of the Polish Mathematical Society. At this session W. Orlicz, president of the Polish Mathematical Society, K.Krickeberg,president of the Bernoulli Society. R. Bartoszynski and K. Doksum gave talks on Neyman IS con­ tribution to statistics, his organizational achievements in the U.S.

Probability has been an important part of mathematics for more than three centuries. Moreover, its importance has grown in recent decades, since the computing power now widely available has allowed probabilistic and stochastic techniques to attack problems such as speech and image processing, geophysical exploration, radar, sonar, etc. -- all of which are covered here. The book contains three exceptionally clear expositions on wavelets, frames and their applications. A further extremely active current research area, well covered here, is the relation between probability and partial differential equations, including probabilistic representations of solutions to elliptic and parabolic PDEs. New approaches, such as the PDE method for large deviation problems, and stochastic optimal control and filtering theory, are beginning to yield their secrets. Another topic dealt with is the application of probabilistic techniques to mathematical analysis. Finally, there are clear explanations of normal numbers and dynamic systems, and the influence of probability on our daily lives.

The papers contained in this volume are an indication of the topics th discussed and the interests of the participants of The 9 International Conference on Probability in Banach Spaces, held at Sandjberg, Denmark, August 16-21, 1993. A glance at the table of contents indicates the broad range of topics covered at this conference. What defines research in this field is not so much the topics considered but the generality of the ques­ tions that are asked. The goal is to examine the behavior of large classes of stochastic processes and to describe it in terms of a few simple prop­ erties that the processes share. The reward of research like this is that occasionally one can gain deep insight, even about familiar processes, by stripping away details, that in hindsight turn out to be extraneous. A good understanding about the disciplines involved in this field can be obtained from the recent book, Probability in Banach Spaces, Springer-Verlag, by M. Ledoux and M. Thlagrand. On page 5, of this book, there is a list of previous conferences in probability in Banach spaces, including the other eight international conferences. One can see that research in this field over the last twenty years has contributed significantly to knowledge in probability and has had important applications in many other branches of mathematics, most notably in statistics and functional analysis.

Probability limit theorems in infinite-dimensional spaces give conditions un­ der which convergence holds uniformly over an infinite class of sets or functions. Early results in this direction were the Glivenko-Cantelli, Kolmogorov-Smirnov and Donsker theorems for empirical distribution functions. Already in these cases there is convergence in Banach spaces that are not only infinite-dimensional but nonsep­ arable. But the theory in such spaces developed slowly until the late 1970's. Meanwhile, work on probability in separable Banach spaces, in relation with the geometry of those spaces, began in the 1950's and developed strongly in the 1960's and 70's. We have in mind here also work on sample continuity and boundedness of Gaussian processes and random methods in harmonic analysis. By the mid-70's a substantial theory was in place, including sharp infinite-dimensional limit theorems under either metric entropy or geometric conditions. Then, modern empirical process theory began to develop, where the collection of half-lines in the line has been replaced by much more general collections of sets in and functions on multidimensional spaces. Many of the main ideas from probability in separable Banach spaces turned out to have one or more useful analogues for empirical processes. Tightness became'asymptotic equicontinuity.'Metric entropy remained useful but also was adapted to metric entropy with bracketing, random entropies, and Kolchinskii-Pollard entropy. Even norms themselves were in some situations replaced by measurable majorants, to which the well-developed separable theory then carried over straightforwardly.

The first international conference on Probability in Banach Spaces was held at Oberwolfach, West Germany, in 1975. It brought together European researchers who, under the inspiration of the Schwartz Seminar in Paris, were using probabi­ listic methods in the study of the geometry of Banach spaces, a rather small number of probabilists who were already studying classical limit laws on Banach spaces, and a larger number of probabilists, specialists in various aspects of the study of Gaussian processes, whose results and techniques were of interest to the members of the first two groups. This first conference was very fruitful. It fos­ tered a continuing relationship among 50 to 75 probabilists and analysts working on probability on infinite-dimensional spaces, the geometry of Banach spaces, and the use of random methods in harmonic analysis. Six more international conferences were held since the 1975 meeting. Two of the meetings were held at Tufts University, one at S¢nderborg, Denmark, and the others at Oberwolfach. This volume contains a selection of papers by the partici­ pants of the Seventh International Conference held at Oberwolfach, West Ger­ many, June 26-July 2, 1988. This exciting and provocative conference was at­ tended by more than 50 mathematicians from many countries. These papers demonstrate the range of interests of the conference participants. In addition to the ongoing study of classical and modern limit theorems in Banach spaces, a branching out has occurred among the members of this group.

This volume contains a selection of papers by the participants of the 6. International Conference on Probability in Banach Spaces, Sand­ bjerg, Denmark, June 16-D1, 1986. The conference was attended by 45 participants from several countries. One thing makes this conference completely different from the previous five ones, namely that it was ar­ ranged jointly in Probability in Banach spaces and Banach space theory with almost equal representation of scientists in the two fields. Though these fields are closely related it seems that direct collaboration between researchers in the two groups has been seldom. It is our feeling that the conference, where the participants were together for five days taking part in lectures and intense discussions of mutual problems, has contributed to a better understanding and closer collaboration in the two fields. The papers in the present volume do not cover all the material pre­ sented in the lectures; several results covered have been published else­ where. The sponsors of the conference are: The Carlsberg Foundation, The Danish Natural Science Research Council, The Danish Department of Education, The Department of Mathematics, Odense University, The Department of Mathematics, Aarhus University, The Knudsen Foundation, Odense, Odense University, The Research Foundation of Aarhus University, The Thborg Foundation. The participants and the organizers would like to thank these institu­ tions for their support. The Organizers. Contents A. de Acosta and M. Ledoux, On the identification of the limits in the law of the iterated logarithm in Banach spaces.....

It is only just recently that people have the tools to judge how well they are doing when making decisions. These tools were conceptualized in the seventeenth century. Since then many people have worked to sharpen the concepts, and to explore how these can be applied further. The problems of decision-making and the theory developed correspondingly have drawn the interest of mathematicians, psychologists, statisticians, economists, philosophers, organizational experts, sociologists, not only for their general relevance, but also for a more intrinsic fascination. There are quite a few institutionalized activities to disseminate results and stimulate research in decision-making. For about a decade now a European organizational structure, centered mainly around the psy­ chological interest in decision-making. There have been conferences in Hamburg, Amsterdam, Uxbridge, Rome and Darmstadt. Conference papers have been partly published+. The organization has thus stabilized, and its re­ latively long history makes it interesting to see what kind of developments occurred, within the area of interest.

Human decision making involves problems which are being studied with increasing interest and sophistication. They range from controversial political decisions via individual consumer decisions to such simple tasks as signal discriminations. Although it would seem that decisions have to do with choices among available actions of any kind, there is general agreement that decision making research should pertain to choice prob­ lems which cannot be solved without a predecisional stage of finding choice alternatives, weighing evidence, and judging values. The ultimate objective of scientific research on decision making is two-fold: (a) to develop a theoretically sound technology for the optimal solution of decision problems, and (b) to formulate a descriptive theory of human decision making. The latter may, in tum, protect decision makers from being caught in the traps of their own limitations and biases. Recently, in decision making research the strong emphasis on well­ defined laboratory tasks is decreasing in favour of more realistic studies in various practical settings. This may well have been caused by a growing awareness of the fact that decision-behaviour is strongly determined by situational factors, which makes it necessary to look into processes of interaction between the decision maker and the relevant task environ­ ment. Almost inevitably there is a parallel shift of interest towards problems of utility measurement and the evaluation of consequences.

This book is based on a number of lectures presented at CISM• -Course on'Stochastic Methods in Structural Mechanics', August 28 -30,1985 in Udine, Italy. The chapters presented here are either expanded and/or updated versions of these lectures. The purpose is to introduce readers to basic principles of stochastic methods of structural mechanics, particularly to those of dynamics. For those readers who wish to pursue the study further, the references provided in each chapter will serve as a useful source of information. Nevertheless the readers find some of the advanced topics presented by the authors immediately useful for their own application. The first section of Chapter 1 introduces the reader to the basic principles of probability theory followed by the discussion of methods to calculate time invariant structural reliability estimates, where the exact methods are particularly emphasized. The Chapter continues with a first introduction to the theory of stochastic processes. The properties of Gaussian and other type of processes are discussed. In dealing with observed data, tests of stationarity, as well as methods to estimate power spectra are described in some detail. The Chapter closes with a first treatice of excursions of stochastic processes in terms of number and duration of excursions, extremes, envelopes and time to first excursions. In Chapter 2 linear structures under stochastic loading are analyzed by applying the concepts as outlined in Chapter 1. The analyses are carried out in the time and frequency range respectively.

Proceedings of a conference held in Santa Barbara, California, May 20-22, 1993

In May of 1973 we organized an international research colloquium on foundations of probability, statistics, and statistical theories of science at the University of Western Ontario. During the past four decades there have been striking formal advances in our understanding of logic, semantics and algebraic structure in probabilistic and statistical theories. These advances, which include the development of the relations between semantics and metamathematics, between logics and algebras and the algebraic-geometrical foundations of statistical theories (especially in the sciences), have led to striking new insights into the formal and conceptual structure of probability and statistical theory and their scientific applications in the form of scientific theory. The foundations of statistics are in a state of profound conflict. Fisher's objections to some aspects of Neyman-Pearson statistics have long been well known. More recently the emergence of Bayesian statistics as a radical alternative to standard views has made the conflict especially acute. In recent years the response of many practising statisticians to the conflict has been an eclectic approach to statistical inference. Many good statisticians have developed a kind of wisdom which enables them to know which problems are most appropriately handled by each of the methods available. The search for principles which would explain why each of the methods works where it does and fails where it does offers a fruitful approach to the controversy over foundations.

On May 27-31, 1985, a series of symposia was held at The University of Western Ontario, London, Canada, to celebrate the 70th birthday of Pro­ fessor V. M. Joshi. These symposia were chosen to reflect Professor Joshi's research interests as well as areas of expertise in statistical science among faculty in the Departments of Statistical and Actuarial Sciences, Economics, Epidemiology and Biostatistics, and Philosophy. From these symposia, the six volumes which comprise the'Joshi Festschrift'have arisen. The 117 articles in this work reflect the broad interests and high quality of research of those who attended our conference. We would like to thank all of the contributors for their superb cooperation in helping us to complete this project. Our deepest gratitude must go to the three people who have spent so much of their time in the past year typing these volumes: Jackie Bell, Lise Constant, and Sandy Tarnowski. This work has been printed from'camera ready'copy produced by our Vax 785 computer and QMS Lasergraphix printers, using the text processing software TEX. At the initiation of this project, we were neophytes in the use of this system. Thank you, Jackie, Lise, and Sandy, for having the persistence and dedication needed to complete this undertaking.

On May 27-31, 1985, a series of symposia was held at The University of Western Ontario, London, Canada, to celebrate the 70th birthday of Pro­ fessor V. M. Joshi. These symposia were chosen to reflect Professor Joshi's research interests as well as areas of expertise in statistical science among faculty in the Departments of Statistical and Actuarial Sciences, Economics, Epidemiology and Biostatistics, and Philosophy. From these symposia, the six volumes which comprise the'Joshi Festschrift'have arisen. The 117 articles in this work reflect the broad interests and high quality of research of those who attended our conference. We would like to thank all of the contributors for their superb cooperation in helping us to complete this project. Our deepest gratitude must go to the three people who have spent so much of their time in the past year typing these volumes: Jackie Bell, Lise Constant, and Sandy Tarnowski. This work has been printed from'carnera ready'copy produced by our Vax 785 computer and QMS Lasergraphix printers, using the text processing software TEX. At the initiation of this project, we were neophytes in the use of this system. Thank you, Jackie, Lise, and Sandy, for having the persistence and dedication needed to complete this undertaking.

The papers collected here are, with three exceptions, those presented at a conference on probability and causation held at the University of California at Irvine on July 15-19, 1985. The exceptions are that David Freedman and Abner Shimony were not able to contribute the papers that they presented to this volume, and that Clark Glymour who was not able to attend the conference did contribute a paper. We would like to thank the National Science Foundation and the School of Humanities of the University of California at Irvine for generous support. WILLIAM HARPER University of Western Ontario BRIAN SKYRMS University of California at Irvine Vll INTRODUCTION PART I: DECISIONS AND GAMES Causal notions have recently corne to figure prominently in discussions about rational decision making. Indeed, a relatively influential new approach to theorizing about rational choice has come to be called'causal decision theory'. 1 Decision problems such as Newcombe's Problem and some versions of the Prisoner's Dilemma where an act counts as evidence for a desired state even though the agent knows his choice of that act cannot causally influence whether or not the state obtains have motivated causal decision theorists.

The papers collected here are, with three exceptions, those presented at a conference on probability and causation held at the University of California at Irvine on July 15-19, 1985. The exceptions are that David Freedman and Abner Shimony were not able to contribute the papers that they presented to this volume, and that Clark Glymour who was not able to attend the conference did contribute a paper. We would like to thank the National Science Foundation and the School of Humanities of the University of California at Irvine for generous support. WILLIAM HARPER University of Western Ontario BRIAN SKYRMS University of California at Irvine VII INTRODUCTION TO CAUSATION, CHANCE, AND CREDENCE The search for causes is so central to science that it has sometimes been taken as the defining attribute of the scientific enterprise. Yet even after twenty-five centuries of philosophical analysis the meaning of'cause'is still a matter of controversy, among scientists as well as philosophers. Part of the problem is that the servicable concepts of causation built out of Necessity, Sufficiency, Locality, and Temporal Precedence were constructed for a deterministic world-view which has been obsolete since the advent of quantum theory. A physically credible theory of causation must be, at basis, statistical. And statistical analyses of caus­ ation may be of interest even when an underlying deterministic theory is assumed, as in classical statistical mechanics.

This volume contains articles from invited speakers at a meeting which took place in Delphi, during the week of October 12-16, 1987. The theme of the meeting was''The concept of probability'and was organized by the'Group of Interdisciplinary Research'(Physics Department, University of Athens) and the Theoretical and Physical Chemistry Institute of the National Hellenic Research Foundation, Athens. (The Group ofInterdisciplinary Research organized two previous Meetings, 1) on the Concept of physical reality (1982) and 2) on the question of determinism in Physics (1984». This small gathering, which was attended by scientists, mathematicians and philosophers from more than 22 countries, took place on the occasion of the 100th year from the birthday of E.Schrodinger. As the father of wave-mechanics, Schrodinger thrushed us into an era of physics where knowledge of the lV-function is considered, for most situations, as the ultimate aim and the ultimate truth. Yet, he, as well as another towering figure of 20th century physics, A.Einstein, never really felt confortable with the interpretation of the meaning oflV and of the information that it contains. With Einstein playing the leading role a debate about concepts and interpretation started as soon as quantum mechanics was born. Central theme to this debate is the concept of probability, a concept which permeates-explicitly or implicitly-all science and even our decision making in everyday life. The articles cover a broad spectrum of thought and results -mathematical, physical, epistemological, experimental, specific, general,-many of them outside the accepted norm.

High dimensional probability, in the sense that encompasses the topics rep­ resented in this volume, began about thirty years ago with research in two related areas: limit theorems for sums of independent Banach space valued random vectors and general Gaussian processes. An important feature in these past research studies has been the fact that they highlighted the es­ sential probabilistic nature of the problems considered. In part, this was because, by working on a general Banach space, one had to discard the extra, and often extraneous, structure imposed by random variables taking values in a Euclidean space, or by processes being indexed by sets in R or Rd. Doing this led to striking advances, particularly in Gaussian process theory. It also led to the creation or introduction of powerful new tools, such as randomization, decoupling, moment and exponential inequalities, chaining, isoperimetry and concentration of measure, which apply to areas well beyond those for which they were created. The general theory of em­ pirical processes, with its vast applications in statistics, the study of local times of Markov processes, certain problems in harmonic analysis, and the general theory of stochastic processes are just several of the broad areas in which Gaussian process techniques and techniques from probability in Banach spaces have made a substantial impact. Parallel to this work on probability in Banach spaces, classical proba­ bility and empirical process theory were enriched by the development of powerful results in strong approximations.

The Athens Conference on Applied Probability and Time Series in 1995 brought together researchers from across the world. The published papers appear in two volumes. Volume I includes papers on applied probability in Honor of J.M. Gani. The topics include probability and probabilistic methods in recursive algorithms and stochastic models, Markov and other stochastic models such as Markov chains, branching processes and semi-Markov systems, biomathematical and genetic models, epidemilogical models including S-I-R (Susceptible-Infective-Removal), household and AIDS epidemics, financial models for option pricing and optimization problems, random walks, queues and their waiting times, and spatial models for earthquakes and inference on spatial models.

Most industrial and natural materials exhibit a macroscopic behaviour which results from the existence of microscale inhomogeneities. The influence of such inhomogeneities is commonly modelled using probabilistic methods. Most of the approaches to the evaluation of the safety of structures according to probabilistic criteria are somewhat scattered, however, and it is time to present such material in a coherent and up-to-date form. Probabilities and Materials undertakes this task, and also defines the great tasks that must be tackled in coming years. For engineers and researchers dealing with materials, geotechnics, solid mechanics, soil mechanics, statistics and stochastic processes. The expository nature of the book means that no prior knowledge of statistics or probability is required of the reader. The book can thus serve as an excellent introduction to the nature of applied statistics and stochastic modelling.

Physicists, when modelling physical systems with a large number of degrees of freedom, and statisticians, when performing data analysis, have developed their own concepts and methods for making the `best'inference. But are these methods equivalent, or not? What is the state of the art in making inferences? The physicists want answers. More: neural computation demands a clearer understanding of how neural systems make inferences; the theory of chaotic nonlinear systems as applied to time series analysis could profit from the experience already booked by the statisticians; and finally, there is a long-standing conjecture that some of the puzzles of quantum mechanics are due to our incomplete understanding of how we make inferences. Matter enough to stimulate the writing of such a book as the present one. But other considerations also arise, such as the maximum entropy method and Bayesian inference, information theory and the minimum description length. Finally, it is pointed out that an understanding of human inference may require input from psychologists. This lively debate, which is of acute current interest, is well summarized in the present work.

The Conference was organized by the Institute of Information Theory and Automation of the Czechoslovak Academy of Sciences from July 7 - 11, 1986, in Prague. The round number of the conference was only one of the jubilees connected with its organization. Namely, thirty years of the Prague Conferences (the first one was organized in autumn 1956 in Liblice near Prague), and two anniversaries of Professor Anton1n Spacek, the inspirator and first organizer of the Prague Conferences - 75 years of his birth and 25 years of his untimely death. (More about Professor Spacek can be found in the Transactions of the Sixth Prague Conferen­ ce) • The Tenth Prague Conference kept the traditional style and orien­ tation typical for the previous Prague Conferences. Almost two hund­ red of participants from 23 countries (Algerie, Austria, Bulgaria, Canada, Czechoslovakia, Federal Republic of Germany, Finland, France, German Democratic Republic, Great Britain, Hungary, Iran, Italy, Ja­ pan, Netherlands, Poland, Roumania, Soviet Union, Sweden, Switzerland, United States, Vietnam and West Berlin) took part in its sessions and discussions. There were 14 invited lectures and. 92 short contri­ butions included in four parallel sections of the Conference program­ me: further, 12 contributions were presented as posters. The invited lectures and submitted contributions covered the three traditional subjects of the Prague Conferences introduced in their title, as well as lots of further applications of the probability theory and mathe­ matical statistics. Most of the presented lectures and contributions are published in the present Conference Transactions.

The Entretiens of the Institut International de Philosophie for 1978 were held in connection with the World Congress of Philosophy in Dusseldorf, from August 27 to September 1. The theme of the Entretiens was Logic and Philosophy (Logique et philosophie). The undersigned, then President of LI.P., was responsible for the planning of the programme. The programme was designed to consist of four sections with the headings Classical and Intuitionist Logic, Modal Logic and its Applications, Inductive Logic and its Applications, and Logic and Epistemology. The aim was also to convey to philosophers who are not experts in logic an informative and representative impression of some of the main sectors of the vast and rapidly expanding field of philosophical logic. At the same time it was thought that this impression should not be conveyed in the form of a series of survey papers but through presentations and discussions of specific topics falling under the main headings men­ tioned above. For each section a rapporteur was nominated to read a paper and an interlocuteur to comment on it. The programme chairman is grateful that he was able to engage a representative selection of front rank philosophi­ cal logicians to perform the various tasks. The papers and the comments are printed in this volume in the order in which they appeared in the Programme of the Entretiens.

Discrete probability theory and the theory of algorithms have become close partners over the last ten years, though the roots of this partnership go back much longer. The papers in this volume address the latest developments in this active field. They are from the IMA Workshops'Probability and Algorithms'and'The Finite Markov Chain Renaissance.'They represent the current thinking of many of the world's leading experts in the field. Researchers and graduate students in probability, computer science, combinatorics, and optimization theory will all be interested in this collection of articles. The techniques developed and surveyed in this volume are still undergoing rapid development, and many of the articles of the collection offer an expositionally pleasant entree into a research area of growing importance.

The past several years have seen the creation and extension of a very conclusive theory of statistics and probability. Many of the research workers who have been concerned with both probability and statistics felt the need for meetings that provide an opportunity for personal con­ tacts among scholars whose fields of specialization cover broad spectra in both statistics and probability: to discuss major open problems and new solutions, and to provide encouragement for further research through the lectures of carefully selected scholars, moreover to introduce to younger colleagues the latest research techniques and thus to stimulate their interest in research. To meet these goals, the series of Pannonian Symposia on Mathematical Statistics was organized, beginning in the year 1979: the first, second and fourth one in Bad Tatzmannsdorf, Burgenland, Austria, the third and fifth in Visegrad, Hungary. The Sixth Pannonian Symposium was held in Bad Tatzmannsdorf again, in the time between 14 and 20 September 1986, under the auspices of Dr. Heinz FISCHER, Federal Minister of Science and Research, Theodor KERY, President of the State Government of Burgenland, Dr. Franz SAUERZOPF, Vice-President of the State Govern­ ment of Burgenland and Dr. Josef SCHMIDL, President of the Austrian Sta­ tistical Central Office. The members of the Honorary Committee were Pal ERDOS, WXadisXaw ORLICZ, Pal REVESz, Leopold SCHMETTERER and Istvan VINCZE; those of the Organizing Committee were Wilfried GROSSMANN (Uni­ versity of Vienna), Franz KONECNY (University of Agriculture of Vienna) and, as the chairman, Wolfgang WERTZ (Technical University of Vienna).

The Fourth International Congress for Logic, Methodology, and Philos­ ophy of Science was held in Bucharest, Romania, on August 29-September 4, 1971. The Congress was organized, under the auspices of the Inter­ national Union for History and Philosophy of Science, Division of Logic, Methodology and Philosophy of Science, by the Academy of the Socialist Republic of Romania, the Academy of Social and Political Sciences of the Socialist Republic of Romania, and the Ministry of Education of Romania. With more than eight hundred participating scholars from thirty-four countries, the Congress was one of the major scientific events of the year 1971. The dedicated efforts of the organizers, the rich and carefully planned program, and the warm and friendly atmosphere contributed to making the Congress a successful and fruitful forum of exchange of scientific ideas. The work of the Congress consisted of invited one hour and half-hour addresses, symposia, and contributed papers. The proceedings were organized into twelve sections of Mathematical Logic, Foundations of Mathematical Theories, Automata and Programming Languages, Philos­ ophy of Logic and Mathematics, General Problems of Methodology and Philosophy of Science, Foundations of Probability and Induction, Methodology and Philosophy of Physical Sciences, Methodology and Philosophy of Biological Sciences, Methodology and Philosophy of Psychological Sciences, Methodology and Philosophy of Historical and Social Sciences, Methodology and Philosophy of Linguistics, and History of Logic, Methodology and Philosophy of Science.