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The notions of transfer function and characteristic functions proved to be fundamental in the last fifty years in operator theory and in system theory. Moshe Livsic played a central role in developing these notions, and the book contains a selection of carefully chosen refereed papers dedicated to his memory. Topics include classical operator theory, ergodic theory and stochastic processes, geometry of smooth mappings, mathematical physics, Schur analysis and system theory. The variety of topics attests well to the breadth of Moshe Livsic's mathematical vision and the deep impact of his work.

Characteristic functions. --- Light -- Scattering. --- Scattering (Physics). --- Transfer functions. --- Mathematics --- Calculus --- Mathematical Statistics --- Physical Sciences & Mathematics --- Characteristic functions --- Scattering (Mathematics) --- Transfer functions --- Operator theory --- Ergodic theory --- Functions, Transfer --- Scattering theory (Mathematics) --- Characteristic formula of an ideal --- Characteristic Hilbert functions --- Functions, Characteristic --- Functions, Hilbert --- Hilbert characteristic functions --- Hilbert functions --- Hilbert's characteristic functions --- Hilbert's functions --- Postulation formula --- Mathematics. --- Operator theory. --- Operator Theory. --- Probabilities --- Automatic control --- Control theory --- Boundary value problems --- Differential equations, Partial --- Scattering operator --- Functional analysis

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This is a text encompassing all of the standard topics in introductory probability theory, together with a significant amount of optional material of emerging importance. The emphasis is on a lucid and accessible writing style, mixed with a large number of interesting examples of a diverse nature. The text will prepare students extremely well for courses in more advanced probability and in statistical theory and for the actuary exam. The book covers combinatorial probability, all the standard univariate discrete and continuous distributions, joint and conditional distributions in the bivariate and the multivariate case, the bivariate normal distribution, moment generating functions, various probability inequalities, the central limit theorem and the laws of large numbers, and the distribution theory of order statistics. In addition, the book gives a complete and accessible treatment of finite Markov chains, and a treatment of modern urn models and statistical genetics. It includes 303 worked out examples and 810 exercises, including a large compendium of supplementary exercises for exam preparation and additional homework. Each chapter has a detailed chapter summary. The appendix includes the important formulas for the distributions in common use and important formulas from calculus, algebra, trigonometry, and geometry. Anirban DasGupta is Professor of Statistics at Purdue University, USA. He has been the main editor of the Lecture Notes and Monographs series, as well as the Collections series of the Institute of Mathematical Statistics, and is currently the Co-editor of the Selected Works in Statistics and Probability series, published by Springer. He has been an associate editor of the Annals of Statistics, Journal of the American Statistical Association, Journal of Statistical Planning and Inference, International Statistical Review, Sankhya, and Metrika. He is the author of Asymptotic Theory of Statistics and Probability, 2008, and of 70 refereed articles on probability and statistics. He is a Fellow of the Institute of Mathematical Statistics.

Electronic books. -- local. --- Probabilities. --- Probabilities --- Mathematics --- Physical Sciences & Mathematics --- Mathematical Statistics --- Probability --- Statistical inference --- Mathematics. --- Probability Theory and Stochastic Processes. --- Combinations --- Chance --- Least squares --- Mathematical statistics --- Risk --- Math --- Science --- Distribution (Probability theory. --- Distribution functions --- Frequency distribution --- Characteristic functions

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This is the first volume of a subseries of the Lecture Notes in Mathematics which will appear randomly over the next years. Each volume will describe some important topic in the theory or applications of Lévy processes and pay tribute to the state of the art of this rapidly evolving subject with special emphasis on the non-Brownian world. The three expository articles of this first volume have been chosen to reflect the breadth of the area of Lévy processes. The first article by Ken-iti Sato characterizes extensions of the class of selfdecomposable distributions on R^d. The second article by Thomas Duquesne discusses Hausdorff and packing measures of stable trees. The third article by Oleg Reichmann and Christoph Schwab presents numerical solutions to Kolmogoroff equations, which arise for instance in financial engineering, when Lévy or additive processes model the dynamics of the risky assets.

Lâevy processes --- Branching processes --- Trees (Graph theory) --- Options (Finance) --- Mathematics --- Physical Sciences & Mathematics --- Mathematical Statistics --- Prices --- Mathematical models --- Branching processes. --- Lévy processes. --- Lévy, Paul, --- Processes, Branching --- Mathematics. --- Probabilities. --- Probability Theory and Stochastic Processes. --- Probability --- Statistical inference --- Combinations --- Chance --- Least squares --- Mathematical statistics --- Risk --- Math --- Science --- Graph theory --- Random walks (Mathematics) --- Stochastic processes --- Distribution (Probability theory. --- Distribution functions --- Frequency distribution --- Characteristic functions --- Probabilities

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Three centuries ago Montmort and De Moivre published two of the first books on probability theory, then called the doctrine of chances, emphasizing its most important application at that time, games of chance. This volume, on the probabilistic aspects of gambling, is a modern version of those classics. While covering the classical material such as house advantage and gambler's ruin, it also takes up such 20th-century topics as martingales, Markov chains, game theory, bold play, and optimal proportional play. In addition there is extensive coverage of specific casino games such as roulette, craps, video poker, baccarat, and twenty-one. The volume addresses researchers and graduate students in probability theory, stochastic processes, game theory, operations research, statistics but it is also accessible to undergraduate students, who have had a course in probability.

Gambling systems. --- Games of chance (Mathematics). --- Games of chance (Mathematics) --- Gambling systems --- Mathematics --- Algebra --- Mathematical Statistics --- Physical Sciences & Mathematics --- Betting systems --- Systems, Gambling --- Gambling problem (Mathematics) --- Mathematics. --- Game theory. --- Probabilities. --- Probability Theory and Stochastic Processes. --- Game Theory, Economics, Social and Behav. Sciences. --- Probability --- Statistical inference --- Combinations --- Chance --- Least squares --- Mathematical statistics --- Risk --- Games, Theory of --- Theory of games --- Mathematical models --- Math --- Science --- Probabilities --- Statistics --- Game theory --- Distribution (Probability theory. --- Distribution functions --- Frequency distribution --- Characteristic functions

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This book offers a comprehensive guide to large sample techniques in statistics. More importantly, it focuses on thinking skills rather than just what formulae to use; it provides motivations, and intuition, rather than detailed proofs; it begins with very simple techniques, and connects theory and applications in entertaining ways. The first five chapters review some of the basic techniques, such as the fundamental epsilon-delta arguments, Taylor expansion, different types of convergence, and inequalities. The next five chapters discuss limit theorems in specific situations of observational data. Each of the first 10 chapters contains at least one section of case study. The last five chapters are devoted to special areas of applications. The sections of case studies and chapters of applications fully demonstrate how to use methods developed from large sample theory in various, less-than-textbook situations. The book is supplemented by a large number of exercises, giving the readers plenty of opportunities to practice what they have learned. The book is mostly self-contained with the appendices providing some backgrounds for matrix algebra and mathematical statistics. The book is intended for a wide audience, ranging from senior undergraduate students to researchers with Ph.D. degrees. A first course in mathematical statistics and a course in calculus are prerequisites. Jiming Jiang is a Professor of Statistics at the University of California, Davis. He is a Fellow of the American Statistical Association and a Fellow of the Institute of Mathematical Statistics. He is the author of another Springer book, Linear and Generalized Linear Mixed Models and Their Applications (2007). Jiming Jiang is a prominent researcher in the fields of mixed effects models, small area estimation and model selection. Most of his research papers have involved large sample techniques. He is currently an Associate Editor of the Annals of Statistics.

Electronic books. -- local. --- Statistics -- Data processing. --- Sampling (Statistics) --- Law of large numbers --- Mathematical statistics --- Mathematics --- Physical Sciences & Mathematics --- Mathematical Statistics --- Statistics --- Data processing. --- Mathematics. --- Probabilities. --- Statistics. --- Probability Theory and Stochastic Processes. --- Statistical Theory and Methods. --- Statistical analysis --- Statistical data --- Statistical methods --- Statistical science --- Econometrics --- Probability --- Statistical inference --- Combinations --- Chance --- Least squares --- Risk --- Math --- Science --- Distribution (Probability theory. --- Mathematical statistics. --- Distribution functions --- Frequency distribution --- Characteristic functions --- Probabilities --- Statistics, Mathematical --- Law of large numbers. --- Statistics .