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• Jorgen Hoffmann-Jorgensen (1942-2017).- Moment estimation implied by the Bobkov-Ledoux inequality.- Polar Isoperimetry. I: The case of the Plane.- Iterated Jackknives and Two-Sided Variance Inequalities.- A Probabilistic Characterization of Negative Definite Functions.- Higher Order Concentration in presence of Poincare-type inequalities.- Rearrangement and Prekopa-Leindler Type Inequalities.- Generalized Semimodularity: Order Statistics.- Geometry of np -Balls: Classical Results and Recent Developments.- Remarks on Superconcentration and Gamma calculus. Application to Spin Glasses.- Asymptotic behavior of Renyi entropy in the Central Limit Theorem.- Uniform-in-Bandwidth Functional Limit Laws for Multivariate Empirical Processes.- Universality of Limiting Spectral Distribution Under Projective Criteria.- Exchangeable Pairs on Wiener Chaos.- Permanental Processes with Kernels That Are Not Equivalent to a Symmetric Matrix.- Pointwise Properties of Martingales with Values in Banach Function Spaces.- Concentration Inequalities for Randomly Permuted Sums.- Uncertainty Quantification for Matrix Compressed Sensing and Quantum Tomography Problems.- Uniform in Bandwidth Estimation of the Gradient Lines of a Density.
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This volume collects selected papers from the 8th High Dimensional Probability meeting held at Casa Matematica 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.
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• Introduction Set-up and main results Local laws for large random matrices Existence, uniqueness and $\mathrm{L}^{2}$-bound Properties of solution Uniform bounds Regularity of solution Perturbations when generating density is small Behavior of generating density where it is small Stability around small minima of generating density Examples Appendix A. Bibliography.
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The authors consider the nonlinear equation $-\frac 1m=z+Sm$ with a parameter $z$ in the complex upper half plane $\mathbb H $, where $S$ is a positivity preserving symmetric linear operator acting on bounded functions. The solution with values in $ \mathbb H$ is unique and its $z$-dependence is conveniently described as the Stieltjes transforms of a family of measures $v$ on $\mathbb R$. In a previous paper the authors qualitatively identified the possible singular behaviors of $v$: under suitable conditions on $S$ we showed that in the density of $v$ only algebraic singularities of degree two or three may occur. In this paper the authors give a comprehensive analysis of these singularities with uniform quantitative controls. They also find a universal shape describing the transition regime between the square root and cubic root singularities. Finally, motivated by random matrix applications in the authors' companion paper they present a complete stability analysis of the equation for any $z\in \mathbb H$, including the vicinity of the singularities.
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• Queueing Models and Stochastic Networks.- Reliability, Risk, Insurance.- Analytical Methods and Limit Theorems.
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This book constitutes the refereed proceedings of the First International Conference on Analytical and Computational Methods in Probability Theory and its Applications, ACMPT 2017, held in Moscow, Russia, in October 2017. The 42 full papers presented were carefully reviewed and selected from 173 submissions. The conference program consisted of four main themes associated with significant contributions made by A.D.Soloviev. These are: Analytical methods in probability theory, Computational methods in probability theory, Asymptotical methods in probability theory, the history of mathematics.
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• 1 The Mathematics of Elections 1.1 The Basic Elements of an Election 1.2 The Plurality Method 1.3 The Borda Count Method 1.4 The Plurality-with-Elimination Method 1.5 The Method of Pairwise Comparisons 1.6 Fairness Criteria and Arrow's Impossibility Theorem
• 2 The Mathematics of Power 2.1 An Introduction to Weighted Voting 2.2 Banzhaf Power 2.3 Shapley-Shubik Power 2.4 Subsets and Permutations
• 3 The Mathematics of Sharing 3.1 Fair-Division Games 3.2 The Divider-Chooser Method 3.3 The Lone-Divider Method 3.4 The Lone-Chooser Method 3.5 The Method of Sealed Bids 3.6 The Method of Markers
• 4 The Mathematics of Apportionment 4.1 Apportionment Problems and Apportionment Methods 4.2 Hamilton's Method 4.3 Jefferson's Method 4.4 Adams's and Webster's Methods 4.5 The Huntington-Hill Method 4.6 The Quota Rule and Apportionment Paradoxes
• 5 The Mathematics of Getting Around 5.1 Street-Routing Problems 5.2 An Introduction to Graphs 5.3 Euler's Theorems and Fleury's Algorithm 5.4 Eulerizing and Semi-Eulerizing Graphs
• 6 The Mathematics of Touring 6.1 What Is a Traveling Salesman Problem? 6.2 Hamilton Paths and Circuits 6.3 The Brute-Force Algorithm 6.4 The Nearest-Neighbor and Repetitive Nearest-Neighbor Algorithms 6.5 The Cheapest-Link Algorithm
• 7 The Mathematics of Networks 7.1 Networks and Trees 7.2 Spanning Trees, MSTs, and MaxSTs 7.3 Kruskal's Algorithm
• 8 The Mathematics of Scheduling 8.1 An Introduction to Scheduling 8.2 Directed Graphs 8.3 Priority-List Scheduling 8.4 The Decreasing-Time Algorithm 8.5 Critical Paths and the Critical-Path Algorithm
• 9 Population Growth Models 9.1 Sequences and Population Sequences 9.2 The Linear Growth Model 9.3 The Exponential Growth Model 9.4 The Logistic Growth Model
• 10 Financial Mathematics 10.1 Percentages 10.2 Simple Interest 10.3 Compound Interest 10.4 Retirement Savings 10.5 Consumer Debt
• 11 The Mathematics of Symmetry 11.1 Rigid Motions 11.2 Reflections 11.3 Rotations 11.4 Translations 11.5 Glide Reflections 11.6 Symmetries and Symmetry Types 11.7 Patterns
• 12 Fractal Geometry 12.1 The Koch Snowflake and Self-Similarity 12.2 The Sierpinski Gasket and the Chaos Game 12.3 The Twisted Sierpinski Gasket 12.4 The Mandelbrot Set
• 13 Fibonacci Numbers and the Golden Ratio 13.1 Fibonacci Numbers 13.2 The Golden Ratio 13.3 Gnomons 13.4 Spiral Growth in Nature
• 14 Censuses, Surveys, Polls, and Studies 14.1 Enumeration 14.2 Measurement 14.3 Cause and Effect
• 15 Graphs, Charts, and Numbers 15.1 Graphs and Charts 15.2 Means, Medians, and Percentiles 15.3 Ranges and Standard Deviations
• 16 Probabilities, Odds, and Expectations 16.1 Sample Spaces and Events 16.2 The Multiplication Rule, Permutations, and Combinations 16.3 Probabilities and Odds 16.4 Expectations 16.5 Measuring Risk
• 17 The Mathematics of Normality 17.1 Approximately Normal Data Sets 17.2 Normal Curves and Normal Distributions 17.3 Modeling Approximately Normal Distributions 17.4 Normality in Random Events Answers to Selected Exercises Index Photo Credits.
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For courses in Liberal Arts Mathematics. Math: Applicable, Accessible, Modern Excursions in Modern Mathematics introduces readers to the power and beauty of math. By developing an appreciation for the aesthetics and applicability of mathematics, readers who previously felt math was an "unknowable" subject can approach it with a new perspective. Contemporary topics ranging from elections, to networks, to analyzing data show readers that math is an accessible tool that can be applicable and interesting for anyone. Refinement and updating of examples and exercises, plus increased resources, makes the 9th Edition a relevant, accessible, and complete program. Also available with MyLab Math. MyLab(TM) Math is an online homework, tutorial, and assessment program designed to work with this text to engage students and improve results. Within its structured environment, students practice what they learn, test their understanding, and pursue a personalized study plan that helps them absorb course material and understand difficult concepts. NOTE: You are purchasing a standalone product; MyLab Math does not come packaged with this content. If you would like to purchase both the physical text and MyLab Math, search for: 0134442229 / 9780134442228 Excursions in Modern Mathematics plus MyLab Math -- Access Card Package Package consists of: *0134468376 / 9780134468372 Excursions in Modern Mathematics*0321431308 / 9780321431301 MyLab Math -- Glue-in Access Card*0321654064 / 9780321654069 MyLab Math Inside Star Sticker MyLab Math should only be purchased when required by an instructor.
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• Combinatorial analysis
• Axioms of probability
• Conditional probability and independence
• Random variables
• Continuous random variables
• Jointly distributed random variables
• Properties of expectation
• Limit theorems
• Additional topics in probability
• Simulation
• Common discrete distributions
• Common continuous distributions.

For upper-level to graduate courses in Probability or Probability and Statistics, for majors in mathematics, statistics, engineering, and the sciences. Explores both the mathematics and the many potential applications of probability theory A First Course in Probability is an elementary introduction to the theory of probability for students in mathematics, statistics, engineering, and the sciences. Through clear and intuitive explanations, it presents not only the mathematics of probability theory, but also the many diverse possible applications of this subject through numerous examples. The 10th Edition includes many new and updated problems, exercises, and text material chosen both for interest level and for use in building student intuition about probability. 0134753119 / 9780134753119 A First Course in Probability, 10/e.
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• Preface--
• 1. Euclidean sections--
• 2. Separable Banach spaces without the approximation property--
• 3. Gaussian processes--
• 4. Reflexive subspaces of L1--
• 5. The method of selectors. Examples of its use--
• 6. The Pisier space of almost surely continuous functions. Applications-- Appendix. News in the theory of infinite-dimensional Banach spaces in the past twenty years G. Godefroy-- An update on some problems in high dimensional convex geometry and related probabilistic results O. Guedon-- A few updates and pointers G. Pisier-- On the mesh condition for Sidon sets L. Rodriguez-Piazza-- Bibliography-- Author index-- Notation index-- Subject index.
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• Preface-- Preliminary chapter--
• 1. Fundamental notions of probability--
• 2. Bases in Banach spaces--
• 3. Unconditional convergence--
• 4. Banach space valued random variables--
• 5. Type and cotype of Banach spaces. Factorisation through a Hilbert space--
• 6. p-summing operators. Applications--
• 7. Some properties of Lp-spaces--
• 8. The space l1-- Annex. Banach algebras, compact Abelian groups-- Bibliography-- Author index-- Notation index-- Subject index.
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• Volume
• 1: Preface-- Preliminary chapter--
• 1. Fundamental notions of probability--
• 2. Bases in Banach spaces--
• 3. Unconditional convergence--
• 4. Banach space valued random variables--
• 5. Type and cotype of Banach spaces. Factorisation through a Hilbert space--
• 6. p-summing operators. Applications--
• 7. Some properties of Lp-spaces--
• 8. The Space l1-- Annex. Banach algebras, compact abelian groups-- Bibliography-- Author index-- Notation index-- Subject index. Volume
• 2: Preface--
• 1. Euclidean sections--
• 2. Separable Banach spaces without the approximation property--
• 3. Gaussian processes--
• 4. Reflexive subspaces of L1--
• 5. The method of selectors. Examples of its use--
• 6. The Pisier space of almost surely continuous functions. Applications-- Appendix. News in the theory of infinite-dimensional Banach spaces in the past twenty years G. Godefroy-- An update on some problems in high dimensional convex geometry and related probabilistic results O. Guedon-- A few updates and pointers G. Pisier-- On the mesh condition for Sidon sets L. Rodriguez-Piazza-- Bibliography-- Author index-- Notation index-- Subject index.
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This two-volume text provides a complete overview of the theory of Banach spaces, emphasising its interplay with classical and harmonic analysis (particularly Sidon sets) and probability. The authors give a full exposition of all results, as well as numerous exercises and comments to complement the text and aid graduate students in functional analysis. The book will also be an invaluable reference volume for researchers in analysis. Volume 1 covers the basics of Banach space theory, operatory theory in Banach spaces, harmonic analysis and probability. The authors also provide an annex devoted to compact Abelian groups. Volume 2 focuses on applications of the tools presented in the first volume, including Dvoretzky's theorem, spaces without the approximation property, Gaussian processes, and more. In volume 2, four leading experts also provide surveys outlining major developments in the field since the publication of the original French edition.
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• 1. Experiments with random outcomes--
• 2. Conditional probability and independence--
• 3. Random variables--
• 4. Approximations of the binomial distribution--
• 5. Transforms and transformations--
• 6. Joint distribution of random variables--
• 7. Sums and symmetry--
• 8. Expectation and variance in the multivariate setting--
• 9. Tail bounds and limit theorems--
• 10. Conditional distribution-- Appendix A. Things to know from calculus-- Appendix B. Set notation and operations-- Appendix C. Counting-- Appendix D. Sums, products and series-- Appendix E. Table of values for Φ(x)-- Appendix F. Table of common probability distributions.
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This classroom-tested textbook is an introduction to probability theory, with the right balance between mathematical precision, probabilistic intuition, and concrete applications. Introduction to Probability covers the material precisely, while avoiding excessive technical details. After introducing the basic vocabulary of randomness, including events, probabilities, and random variables, the text offers the reader a first glimpse of the major theorems of the subject: the law of large numbers and the central limit theorem. The important probability distributions are introduced organically as they arise from applications. The discrete and continuous sides of probability are treated together to emphasize their similarities. Intended for students with a calculus background, the text teaches not only the nuts and bolts of probability theory and how to solve specific problems, but also why the methods of solution work.
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• Preface-- List of symbols--
• 1. Poisson and other discrete distributions--
• 2. Point processes--
• 3. Poisson processes--
• 4. The Mecke equation and factorial measures--
• 5. Mappings, markings and thinnings--
• 6. Characterisations of the Poisson process--
• 7. Poisson processes on the real line--
• 8. Stationary point processes--
• 9. The Palm distribution--
• 10. Extra heads and balanced allocations--
• 11. Stable allocations--
• 12. Poisson integrals--
• 13. Random measures and Cox processes--
• 14. Permanental processes--
• 15. Compound Poisson processes--
• 16. The Boolean model and the Gilbert graph--
• 17. The Boolean model with general grains--
• 18. Fock space and chaos expansion--
• 19. Perturbation analysis--
• 20. Covariance identities--
• 21. Normal approximation--
• 22. Normal approximation in the Boolean model-- Appendix A. Some measure theory-- Appendix B. Some probability theory-- Appendix C. Historical notes-- References-- Index.
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The Poisson process, a core object in modern probability, enjoys a richer theory than is sometimes appreciated. This volume develops the theory in the setting of a general abstract measure space, establishing basic results and properties as well as certain advanced topics in the stochastic analysis of the Poisson process. Also discussed are applications and related topics in stochastic geometry, including stationary point processes, the Boolean model, the Gilbert graph, stable allocations, and hyperplane processes. Comprehensive, rigorous, and self-contained, this text is ideal for graduate courses or for self-study, with a substantial number of exercises for each chapter. Mathematical prerequisites, mainly a sound knowledge of measure-theoretic probability, are kept in the background, but are reviewed comprehensively in the appendix. The authors are well-known researchers in probability theory; especially stochastic geometry. Their approach is informed both by their research and by their extensive experience in teaching at undergraduate and graduate levels.
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We live in a much more turbulent world than we like to think, but the science we use to analyze economic, financial, and statistical events mostly disregards the world's essentially chaotic nature. We need to get used to the idea that wildly improbable events are actually part of the natural order. The renowned Hungarian mathematician and psychologist L�szl� M�ro explains how the wild and mild worlds (which he names Wildovia and Mildovia) coexist, and that different laws apply to each. Even if we live in an ultimately wild universe, he argues, we're better off pretending that it obeys Mildovian laws. Doing so may amount to a self-fulfilling prophecy and create an island of predictability in a very rough sea. Perched on the ragged border between economics and complexity theory, M�ro proposes to extend the reach of science to subjects previously considered outside its grasp: the unpredictable, unrepeatable, highly improbable events we commonly call "miracles.".
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• 1. Foundations of probability theory--
• 2. Conditional probability--
• 3. Discrete random variables--
• 4. Continuous random variables--
• 5. Jointly distributed random variables--
• 6. Multivariate normal distribution--
• 7. Conditioning by random variables--
• 8. Generating functions--
• 9. Additional topics in probability--
• 10. Discrete-time Markov chains--
• 11. Continuous-time Markov chains.
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Probability has applications in many areas of modern science, not to mention in our daily life. Its importance as a mathematical discipline cannot be overrated, and it is a fascinating and surprising topic in its own right. This engaging textbook with its easy-to-follow writing style provides a comprehensive, yet concise introduction to the subject. It covers all of the standard material for undergraduate and first-year-graduate-level courses as well as many topics that are usually not found in standard text - such as Bayesian inference, Markov chain Monte Carlo simulation, and Chernoff bounds.
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• Preface--
• 1. Random walks on graphs--
• 2. Uniform spanning tree--
• 3. Percolation and self-avoiding walk--
• 4. Association and influence--
• 5. Further percolation--
• 6. Contact process--
• 7. Gibbs states--
• 8. Random-cluster model--
• 9. Quantum Ising model--
• 10. Interacting particle systems--
• 11. Random graphs--
• 12. Lorentz gas-- References-- Index.
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This introduction to some of the principal models in the theory of disordered systems leads the reader through the basics, to the very edge of contemporary research, with the minimum of technical fuss. Topics covered include random walk, percolation, self-avoiding walk, interacting particle systems, uniform spanning tree, random graphs, as well as the Ising, Potts, and random-cluster models for ferromagnetism, and the Lorentz model for motion in a random medium. This new edition features accounts of major recent progress, including the exact value of the connective constant of the hexagonal lattice, and the critical point of the random-cluster model on the square lattice. The choice of topics is strongly motivated by modern applications, and focuses on areas that merit further research. Accessible to a wide audience of mathematicians and physicists, this book can be used as a graduate course text. Each chapter ends with a range of exercises.
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• Preface-- Notation--
• 1. Introduction to combinatorics--
• 2. Probabilistic Moments and Cumulants--
• 3. Quantum probability--
• 4. Quantum fields--
• 5. Combinatorial species--
• 6. Combinatorial aspects of quantum fields: Feynman diagrams--
• 7. Entropy, large deviations and legendre transforms--
• 8. Introduction to Fock spaces--
• 9. Operators and fields on the Boson Fock space--
• 10. L2-representations of the Boson Fock space--
• 11. Local fields on the Boson Fock space: free fields--
• 12. Local fields on the Boson Fock space: interacting fields--
• 13. Quantum stochastic calculus--
• 14. Quantum stochastic limits-- Bibliography-- Index.
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Wick ordering of creation and annihilation operators is of fundamental importance for computing averages and correlations in quantum field theory and, by extension, in the Hudson-Parthasarathy theory of quantum stochastic processes, quantum mechanics, stochastic processes, and probability. This book develops the unified combinatorial framework behind these examples, starting with the simplest mathematically, and working up to the Fock space setting for quantum fields. Emphasizing ideas from combinatorics such as the role of lattice of partitions for multiple stochastic integrals by Wallstrom-Rota and combinatorial species by Joyal, it presents insights coming from quantum probability. It also introduces a 'field calculus' which acts as a succinct alternative to standard Feynman diagrams and formulates quantum field theory (cumulant moments, Dyson-Schwinger equation, tree expansions, 1-particle irreducibility) in this language. Featuring many worked examples, the book is aimed at mathematical physicists, quantum field theorists, and probabilists, including graduate and advanced undergraduate students.
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• - Ornstein-Uhlenbeck Pinball and the Poincare Inequality in a Punctured Domain. - A Probabilistic Look at Conservative Growth-Fragmentation Equations. - Iterated Proportional Fitting Procedure and Infinite Products of Stochastic Matrices. - Limiting Eigenvectors of Outliers for Spiked Information-Plus-Noise Type Matrices. - Criteria for Exponential Convergence to Quasi-Stationary Distributions and Applications to Multi-Dimensional Diffusions. - Bismut-Elworthy-Li Formulae for Bessel Processes. - Large Deviations for Infectious Diseases Models. - The Girsanov Theorem Without (So Much) Stochastic Analysis. - On Drifting Brownian Motion Made Periodic. - On the Markovian Similarity. - Sharp Rate for the Dual Quantization Problem. - Cramer's Theorem in Banach Spaces Revisited. - On Martingale Chaoses. - Explicit Laws for the Records of the Perturbed Random Walk on Z. - A Potential Theoretic Approach to Tanaka Formula for Asymmetric Levy Processes.
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This 49th volume offers a good sample of the main streams of current research on probability and stochastic processes, in particular those active in France. This includes articles on latest developments on diffusion processes, large deviations, martingale theory, quasi-stationary distribution, random matrices, and many more. All the contributions come from spontaneous submissions and their diversity illustrates the good health of this branch of mathematics. The featured contributors are E. Boissard, F. Bouguet, J. Brossard, M. Capitaine, P. Cattiaux, N. Champagnat, K. Abdoulaye Coulibaly-Pasquier, H. Elad Altman, A. Guillin, P. Kratz, A. Lejay, C. Leuridan, P. McGill, L. Miclo, G. Pages, E. Pardoux, P. Petit, B. Rajeev, L. Serlet, H. Tsukada, D. Villeomannais and B. Wilbertz.
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• Preface ixAcknowledgments xi1 Measurement 12 Judgment 223 Psychology 484 Frequency 625 Mathematics 796 Inverse Inference 1007 Unification 1228 Algorithmic Randomness 1459 Physical Chance 16510 Induction 190Appendix: Probability Tutorial 209Notes 225Annotated Select Bibliography 239Image Credits 247Index 249.
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A fascinating account of the breakthrough ideas that transformed probability and statistics In the sixteenth and seventeenth centuries, gamblers and mathematicians transformed the idea of chance from a mystery into the discipline of probability, setting the stage for a series of breakthroughs that enabled or transformed innumerable fields, from gambling, mathematics, statistics, economics, and finance to physics and computer science. This book tells the story of ten great ideas about chance and the thinkers who developed them, tracing the philosophical implications of these ideas as well as their mathematical impact. Persi Diaconis and Brian Skyrms begin with Girolamo Cardano, a sixteenth-century physician, mathematician, and professional gambler who helped develop the idea that chance actually can be measured. They describe how later thinkers showed how the judgment of chance also can be measured, how frequency is related to chance, and how chance, judgment, and frequency could be unified. Diaconis and Skyrms explain how Thomas Bayes laid the foundation of modern statistics, and they explore David Hume's problem of induction, Andrey Kolmogorov's general mathematical framework for probability, the application of computability to chance, and why chance is essential to modern physics. A final idea--that we are psychologically predisposed to error when judging chance--is taken up through the work of Daniel Kahneman and Amos Tversky. Complete with a brief probability refresher, Ten Great Ideas about Chance is certain to be a hit with anyone who wants to understand the secrets of probability and how they were discovered.
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• Weak convergence of measures on $\mathbb{R}^d$ Convergence of measures on metric spaces Metrics on spaces of measures Convergence of measures on topological spaces Spaces of measures with the weak topology Comments Bibliography Index.
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This book provides a thorough exposition of the main concepts and results related to various types of convergence of measures arising in measure theory, probability theory, functional analysis, partial differential equations, mathematical physics, and other theoretical and applied fields. Particular attention is given to weak convergence of measures. The principal material is oriented toward a broad circle of readers dealing with convergence in distribution of random variables and weak convergence of measures. The book contains the necessary background from measure theory and functional analysis. Large complementary sections aimed at researchers present the most important recent achievements. More than 100 exercises (ranging from easy introductory exercises to rather difficult problems for experienced readers) are given with hints, solutions, or references. Historic and bibliographic comments are included. The target readership includes mathematicians and physicists whose research is related to probability theory, mathematical statistics, functional analysis, and mathematical physics.
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• Scaling limits of Markov-Branching trees and applications.- Optimality of two-parameter strategies in stochastic control.- Asymptotic results for the severity and surplus before ruin for a class of Levy insurance processes.- Characterization of the minimal penalty of a convex risk measure with applications to robust utility maximization for Levy models.- Blackwell-Nash equilibria in zero-sum stochastic differential games.- A note on Gamma-convergence of monotone functionals.- A criterion for blow up in finite time of a system of 1-dimensional reaction-diffusion equations.- A note on the small-time behavior of the largest block size of Beta n-coalescents.
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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 Benedicte 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.
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• IntroductionThe time constant and the limit shapeFluctuations and concentration boundsGeodesicsBusemann functionsGrowth and competition modelsVariants of FPP and related modelsSummary of open questionsBibliography.
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First-passage percolation (FPP) is a fundamental model in probability theory that has a wide range of applications to other scientific areas (growth and infection in biology, optimization in computer science, disordered media in physics), as well as other areas of mathematics, including analysis and geometry. FPP was introduced in the 1960s as a random metric space. Although it is simple to define, and despite years of work by leading researchers, many of its central problems remain unsolved.In this book, the authors describe the main results of FPP, with two purposes in mind. First, they give self-contained proofs of seminal results obtained until the 1990s on limit shapes and geodesics. Second, they discuss recent perspectives and directions including (1) tools from metric geometry, (2) applications of concentration of measure, and (3) related growth and competition models. The authors also provide a collection of old and new open questions. This book is intended as a textbook for a graduate course or as a learning tool for researchers.
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• Introduction.- 1.Events and probability.- 2.Random variables.- 3.Bounds and inequalities.- 4.Almost-sure convergence.- 5.Coupling and the variation distance.- 6.The probabilistic method.- 7.Codes and trees.- 8.Markov chains.- 9.Branching trees.- 10.Markov fields on graphs.- 11.Random graphs.- 12.Recurrence of Markov chains.- 13.Random walks on graphs.- 14.Asymptotic behaviour of Markov chains.- 15.Monte Carlo sampling.-
• 16. Convergence rates.- Appendix.- Bibliography.
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The emphasis in this book is placed on general models (Markov chains, random fields, random graphs), universal methods (the probabilistic method, the coupling method, the Stein-Chen method, martingale methods, the method of types) and versatile tools (Chernoff's bound, Hoeffding's inequality, Holley's inequality) whose domain of application extends far beyond the present text. Although the examples treated in the book relate to the possible applications, in the communication and computing sciences, in operations research and in physics, this book is in the first instance concerned with theory. The level of the book is that of a beginning graduate course. It is self-contained, the prerequisites consisting merely of basic calculus (series) and basic linear algebra (matrices). The reader is not assumed to be trained in probability since the first chapters give in considerable detail the background necessary to understand the rest of the book.
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• L. Bunimovich, N. I. Chernov (1956-2014) T. Adams and J. Rosenblatt, Joint coboundaries P. Balint, N. Chernov, and D. Dolgopyat, Convergence of moments for dispersing billiards with cusps E. Catsigeras, M. Cerminara, and H. Enrich, Weak pseudo-physical measures and Pesin's entropy formula for Anosov $C^1$-diffeomorphisms C. Cox and R. Feres, No-slip billiards in dimension two C. P. Dettmann, How sticky is the chaos/order boundary? G. Galperin and M. Levi, Bouncing in gravitational field N. T. A. Haydn and F. Yang, A derivation of the Poisson law for returns of smooth maps with certain geometrical properties K. Khanin and S. Kocic, Rigidity for a class of generalized interval exchange transformations C. C. Moxley and N. J. Simanyi, Homotopical complexity of a $3D$ billiard flow M. Jakobson, Mixing properties of some maps with countable Markov partitions Ya. G. Sinai and I. Vinogradov, Eigenfunctions of Laplacians in some two-dimensional domains D. Szasz, Multidimensional hyperbolic billiards X. Xia and P. Zhang, Homoclinic intersections for geodesic flows on convex spheres H. Zhang, Decay of correlations for billiards with flat points I: Channel effects H. Zhang, Decay of correlations for billiards with flat points II: Cusps effect.
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This volume contains the proceedings of the Conference on Dynamical Systems, Ergodic Theory, and Probability, which was dedicated to the memory of Nikolai Chernov, held from May 18-20, 2015, at the University of Alabama at Birmingham, Birmingham, Alabama. The book is devoted to recent advances in the theory of chaotic and weakly chaotic dynamical systems and its applications to statistical mechanics. The papers present new original results as well as comprehensive surveys.
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In comparison with the first edition of the book, this second edition contains additional topics such as power, sample size computation and non-parametric methods, and includes a large collection of new problems, as well as the answers to odd-numbered problems. Several sections of this edition are accompanied by instructions using the programming language R for statistical computing and graphics.
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