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This is a collection of papers by participants at the 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.

Differential equations, Partial -- Congresses. --- Harmonic analysis -- Congresses. --- Probabilities -- Congresses. --- Stochastic analysis -- Congresses. --- Mathematics --- Physical Sciences & Mathematics --- Mathematical Statistics --- Probabilities. --- Linear topological spaces. --- Topological linear spaces --- Topological vector spaces --- Vector topology --- Probability --- Statistical inference --- Mathematics. --- Computer science --- Computer mathematics. --- Calculus of variations. --- Probability Theory and Stochastic Processes. --- Mathematical Applications in Computer Science. --- Calculus of Variations and Optimal Control; Optimization. --- Combinations --- Chance --- Least squares --- Mathematical statistics --- Risk --- Topology --- Vector spaces --- Distribution (Probability theory. --- Mathematical optimization. --- Optimization (Mathematics) --- Optimization techniques --- Optimization theory --- Systems optimization --- Mathematical analysis --- Maxima and minima --- Operations research --- Simulation methods --- System analysis --- Distribution functions --- Frequency distribution --- Characteristic functions --- Probabilities --- Computer science—Mathematics. --- Isoperimetrical problems --- Variations, Calculus of --- Computer mathematics --- Electronic data processing

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The term High Dimensional Probability in the title of this volume refers to a circle of ideas and problems that originated in Probability in Banach Spaces and the Theory of Gaussian Processes more than forty years ago. Initially, the main focus was on the study of necessary and sufficient conditions for the continuity of Gaussian processes and of classical limit theorems-laws of large numbers, laws of iterated logarithm and central limit theorems in Banach spaces.

Probabilities

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This book focuses on quantitative approximation results for weak limit theorems when the target limiting law is infinitely divisible with finite first moment. Two methods are presented and developed to obtain such quantitative results. At the root of these methods stands a Stein characterizing identity discussed in the third chapter and obtained thanks to a covariance representation of infinitely divisible distributions. The first method is based on characteristic functions and Stein type identities when the involved sequence of random variables is itself infinitely divisible with finite first moment. In particular, based on this technique, quantitative versions of compound Poisson approximation of infinitely divisible distributions are presented. The second method is a general Stein's method approach for univariate selfdecomposable laws with finite first moment. Chapter 6 is concerned with applications and provides general upper bounds to quantify the rate of convergence in classical weak limit theorems for sums of independent random variables. This book is aimed at graduate students and researchers working in probability theory and mathematical statistics.

Approximation theory. --- Theory of approximation --- Functional analysis --- Functions --- Polynomials --- Chebyshev systems --- Distribution (Probability theory. --- Probability Theory and Stochastic Processes. --- Distribution functions --- Frequency distribution --- Characteristic functions --- Probabilities --- Probabilities. --- Probability --- Statistical inference --- Combinations --- Mathematics --- Chance --- Least squares --- Mathematical statistics --- Risk

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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 and perspectives to analyze random phenomena.

Mathematics. --- Probabilities. --- Probability Theory and Stochastic Processes. --- Probabilities --- Distribution (Probability theory. --- Distribution functions --- Frequency distribution --- Characteristic functions --- Probability --- Statistical inference --- Combinations --- Mathematics --- Chance --- Least squares --- Mathematical statistics --- Risk