Probability Distributions on Banach Spaces

Probability Distributions on Banach Spaces

Approach your problems from the right end It isn't that they can't see the solution.

Author: N Vakhania

Publisher: Springer Science & Business Media

ISBN: 9789400938731

Category: Mathematics

Page: 482

View: 243

Approach your problems from the right end It isn't that they can't see the solution. It is and begin with the answers. Then one day, that they can't see the problem. perhaps you will find the final question. G. K. Chesterton. The Scandal of Father 'The Hermit Clad in Crane Feathers' in R Brown 'The point of a Pin'. van Gulik's The Chinese Maze Murders. Growing specialization and diversification have brought a host of monographs and textbooks on increasingly specialized topics. However, the "tree" of knowledge of mathematics and related fields does not grow only by putting forth new branches. It also happens, quite often in fact, that branches which were thought to be completely disparate are suddenly seen to be related. Further, the kind and level of sophistication of mathematics applied in various sciences has changed drastically in recent years: measure theory is used (non trivially) in regional and theoretical economics; algebraic geometry interacts with physics; the Minkowsky lemma, coding theory and the structure of water meet one another in packing and covering theory; quantum fields, crystal defects and mathematical programming profit from homotopy theory; Lie algebras are relevant to filtering; and prediction and electrical engineering can use Stein spaces. And in addition to this there are such new emerging subdisciplines as "experimental mathematics", "CFD", "completely integrable systems", "chaos, synergetics and large-scale order", which are almost impossible to fit into the existing classification schemes. They draw upon widely different sections of mathematics.
Categories: Mathematics

Probability in Banach Spaces

Probability in Banach Spaces

Tbilissi, 1971] N. N. Vakhania, V. I. Tarieladze, S. A. Chobanyan: Probability
distributions on Banach spaces. Reidel, Dordrecht 1987 ... Math. J. 60, 897–936 (
1930) R. Wittmann: A general law of the iterated logarithm. Z.
Wahrscheinlichkeitstheor. Verw. Geb. 68,521—543 ... 5, 283–286 (1977) J. Zinn:
Inequalities in Banach spaces with applications to probabilistic limit theorems: a
survey. Probability in ...

Author: Michel Ledoux

Publisher: Springer Science & Business Media

ISBN: 9783642202124

Category: Mathematics

Page: 480

View: 742

Isoperimetric, measure concentration and random process techniques appear at the basis of the modern understanding of Probability in Banach spaces. Based on these tools, the book presents a complete treatment of the main aspects of Probability in Banach spaces (integrability and limit theorems for vector valued random variables, boundedness and continuity of random processes) and of some of their links to Geometry of Banach spaces (via the type and cotype properties). Its purpose is to present some of the main aspects of this theory, from the foundations to the most important achievements. The main features of the investigation are the systematic use of isoperimetry and concentration of measure and abstract random process techniques (entropy and majorizing measures). Examples of these probabilistic tools and ideas to classical Banach space theory are further developed.
Categories: Mathematics

Introduction to Banach Spaces Analysis and Probability

Introduction to Banach Spaces  Analysis and Probability

This first volume of a two-volume overview covers the basic theory of Banach spaces, harmonic analysis and probability.

Author: Daniel Li

Publisher: Cambridge University Press

ISBN: 9781107160514

Category: Mathematics

Page: 478

View: 336

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.
Categories: Mathematics

Asymptotic Methods in Probability and Statistics with Applications

Asymptotic Methods in Probability and Statistics with Applications

The Laplace method for probability measures in Banach spaces, Russian Math.
... of a Gaussian field with constant variance on a smooth manifold, Probability
Theory and Its Applications, 41, 438-451. 5. ... 14 Typical Distributions: Infinite-
Dimensional Approaches A. V. Sudakov, V. N. Gaussian Non-Centered Fields
203 ...

Author: N. Balakrishnan

Publisher: Springer Science & Business Media

ISBN: 0817642145

Category: Business & Economics

Page: 549

View: 139

This book represents thirty-eight extensive and carefully edited chapters written by prominent researchers, providing an up-to-date survey of new asymptotic methods in science and technology. The chapters contain broad coverage of the latest developments and innovative techniques in a wide range of theoretical and numerical issues in the field of asymptotic methods in probability and mathematical statistics. The book is organized into ten thematic parts: probability distributions; characterizations of distributions; probabilities and measures in high dimensional structures; weak and stron limit theorems; large deviation probabilities; empirical processes; order statistics and records; estimation of parameters and hypotheses testing; random walks, and applications to finance. Written in an accessible style, this book conveys a clear and practical perspective of asymptotic methods. Topics and features:Recent developments in asymptotic methods; Parametric and Nonparametric Inference; Distribution Theory; Stochastic Processes; Order Statistics; Record values and Characterizations. Asymptotic methods in Probability and Mathematical Statistics is an essential resource for reseachers, practitioners, and professionals involved in Theoretical and Applied Probability and/or in Theoretical and Applied Statistics. Various chapters of the volume will also appeal to industrial statisticians and financial economists.
Categories: Business & Economics

High Dimensional Probability VIII

High Dimensional Probability VIII

This volume collects selected papers from the 8th High Dimensional Probability meeting held at Casa Matemática Oaxaca (CMO), Mexico.

Author: Nathael Gozlan

Publisher: Springer Nature

ISBN: 9783030263911

Category: Mathematics

Page: 458

View: 944

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

High Dimensional Probability VI

High Dimensional Probability VI

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.

Author: Christian Houdré

Publisher: Springer Science & Business Media

ISBN: 9783034804905

Category: Mathematics

Page: 374

View: 551

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.​
Categories: Mathematics

Probability in Banach Spaces 6

Probability in Banach Spaces 6

[ 6 ] W . Feller , “ An Introduction to Probability Theory and Its Applications II , ”
Wiley , New York , 1971 . ... Math . Soc . 136 ( 1969 ) , 51 - 65 . ( 14 ) G . Siegel ,
Operator - stable distributions in separable Banach spaces . ( 15 ) N . N .
Vakhania , V ...

Author: Haagerup

Publisher: Birkhäuser

ISBN: 0817634940

Category: Mathematics

Page: 290

View: 906

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. . . . .
Categories: Mathematics

U Statistics in Banach Spaces

U Statistics in Banach Spaces

In this book, the author presents in a systematic form the probabilistic theory of U-statistics with values in Banach spaces (UB-statistics), which has been developed to date.

Author: IU. IUrii Vasilevich Borovskikh

Publisher: VSP

ISBN: 9067642002

Category: Mathematics

Page: 420

View: 264

U-statistics are universal objects of modern probabilistic summation theory. They appear in various statistical problems and have very important applications. The mathematical nature of this class of random variables has a functional character and, therefore, leads to the investigation of probabilistic distributions in infinite-dimensional spaces. The situation when the kernel of a U-statistic takes values in a Banach space, turns out to be the most natural and interesting. In this book, the author presents in a systematic form the probabilistic theory of U-statistics with values in Banach spaces (UB-statistics), which has been developed to date. The exposition of the material in this book is based around the following topics: algebraic and martingale properties of U-statistics; inequalities; law of large numbers; the central limit theorem; weak convergence to a Gaussian chaos and multiple stochastic integrals; invariance principle and functional limit theorems; estimates of the rate of weak convergence; asymptotic expansion of distributions; large deviations; law of iterated logarithm; dependent variables; relation between Banach-valued U-statistics and functionals from permanent random measures.
Categories: Mathematics

Probability Distributions on Linear Spaces

Probability Distributions on Linear Spaces

On the estimation of the convergence rate in the central limit theorem in Hilbert
space ( Russian ) . Trudy VC AN ... On the distribution of the inner product of
Gaussian random vectors ( Russian ) . Soobshtshenia AN ... Theory of Probability
and Its Applications 12 : 525 – 528 . 42 . Prohorov ... Studia Math 29 : 243 – 248 .
47 .

Author: Nikolaĭ Nikolaevich Vakhanii͡a

Publisher: North-Holland

ISBN: UCAL:B4405377

Category: Distribution (Probability theory)

Page: 123

View: 379

Categories: Distribution (Probability theory)

Probability in Banach Spaces 8 Proceedings of the Eighth International Conference

Probability in Banach Spaces  8  Proceedings of the Eighth International Conference

TAIL ESTIMATES FOR EMPIRICAL CHARACTERISTIC FUNCTIONS, WITH
APPLICATIONS TO RANDOM ARRAYS ... the distribution of the supremum of
expressions (1.1) has been investigated in both the mathematical and the
engineering ...

Author: R.M. Dudley

Publisher: Springer Science & Business Media

ISBN: 0817636579

Category: Mathematics

Page: 510

View: 607

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.
Categories: Mathematics

Functional Analysis for Probability and Stochastic Processes

Functional Analysis for Probability and Stochastic Processes

This text is designed both for students of probability and stochastic processes, and for students of functional analysis.

Author: Adam Bobrowski

Publisher: Cambridge University Press

ISBN: 1139443887

Category: Mathematics

Page:

View: 328

This text is designed both for students of probability and stochastic processes, and for students of functional analysis. For the reader not familiar with functional analysis a detailed introduction to necessary notions and facts is provided. However, this is not a straight textbook in functional analysis; rather, it presents some chosen parts of functional analysis that can help understand ideas from probability and stochastic processes. The subjects range from basic Hilbert and Banach spaces, through weak topologies and Banach algebras, to the theory of semigroups of bounded linear operators. Numerous standard and non-standard examples and exercises make the book suitable as a course textbook or for self-study.
Categories: Mathematics

High Dimensional Probability VII

High Dimensional Probability VII

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.

Author: Christian Houdré

Publisher: Birkhäuser

ISBN: 9783319405193

Category: Mathematics

Page: 461

View: 353

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.
Categories: Mathematics

Mathematical Reviews

Mathematical Reviews

Yang, Xin Jian LP integrability of densities of random vectors and its applications.
... summary) 93d:60020 Yanushkevichius, R. On the stability of characterizations
of a mixture of probability distributions. ... 93k:601.36 Siebert, Eberhard Strongly
operator-decomposable probability measures on separable Banach spaces.

Author:

Publisher:

ISBN: UOM:39015051367665

Category: Mathematics

Page:

View: 505

Categories: Mathematics

Measure Theory and Probability Theory

Measure Theory and Probability Theory

This is a graduate level textbook on measure theory and probability theory. It presents the main concepts and results in measure theory and probability theory in a simple and easy-to-understand way.

Author: Krishna B. Athreya

Publisher: Springer Science & Business Media

ISBN: 9780387329031

Category: Business & Economics

Page: 618

View: 760

This is a graduate level textbook on measure theory and probability theory. The book can be used as a text for a two semester sequence of courses in measure theory and probability theory, with an option to include supplemental material on stochastic processes and special topics. It is intended primarily for first year Ph.D. students in mathematics and statistics although mathematically advanced students from engineering and economics would also find the book useful. Prerequisites are kept to the minimal level of an understanding of basic real analysis concepts such as limits, continuity, differentiability, Riemann integration, and convergence of sequences and series. A review of this material is included in the appendix. The book starts with an informal introduction that provides some heuristics into the abstract concepts of measure and integration theory, which are then rigorously developed. The first part of the book can be used for a standard real analysis course for both mathematics and statistics Ph.D. students as it provides full coverage of topics such as the construction of Lebesgue-Stieltjes measures on real line and Euclidean spaces, the basic convergence theorems, L^p spaces, signed measures, Radon-Nikodym theorem, Lebesgue's decomposition theorem and the fundamental theorem of Lebesgue integration on R, product spaces and product measures, and Fubini-Tonelli theorems. It also provides an elementary introduction to Banach and Hilbert spaces, convolutions, Fourier series and Fourier and Plancherel transforms. Thus part I would be particularly useful for students in a typical Statistics Ph.D. program if a separate course on real analysis is not a standard requirement. Part II (chapters 6-13) provides full coverage of standard graduate level probability theory. It starts with Kolmogorov's probability model and Kolmogorov's existence theorem. It then treats thoroughly the laws of large numbers including renewal theory and ergodic theorems with applications and then weak convergence of probability distributions, characteristic functions, the Levy-Cramer continuity theorem and the central limit theorem as well as stable laws. It ends with conditional expectations and conditional probability, and an introduction to the theory of discrete time martingales. Part III (chapters 14-18) provides a modest coverage of discrete time Markov chains with countable and general state spaces, MCMC, continuous time discrete space jump Markov processes, Brownian motion, mixing sequences, bootstrap methods, and branching processes. It could be used for a topics/seminar course or as an introduction to stochastic processes. Krishna B. Athreya is a professor at the departments of mathematics and statistics and a Distinguished Professor in the College of Liberal Arts and Sciences at the Iowa State University. He has been a faculty member at University of Wisconsin, Madison; Indian Institute of Science, Bangalore; Cornell University; and has held visiting appointments in Scandinavia and Australia. He is a fellow of the Institute of Mathematical Statistics USA; a fellow of the Indian Academy of Sciences, Bangalore; an elected member of the International Statistical Institute; and serves on the editorial board of several journals in probability and statistics. Soumendra N. Lahiri is a professor at the department of statistics at the Iowa State University. He is a fellow of the Institute of Mathematical Statistics, a fellow of the American Statistical Association, and an elected member of the International Statistical Institute.
Categories: Business & Economics

Nonlinear Mathematics for Uncertainty and its Applications

Nonlinear Mathematics for Uncertainty and its Applications

This volume is a collection of papers presented at the international conference on Nonlinear Mathematics for Uncertainty and Its Applications (NLMUA2011), held at Beijing University of Technology during the week of September 7--9, 2011.

Author: Shoumei Li

Publisher: Springer Science & Business Media

ISBN: 9783642228339

Category: Computers

Page: 709

View: 170

This volume is a collection of papers presented at the international conference on Nonlinear Mathematics for Uncertainty and Its Applications (NLMUA2011), held at Beijing University of Technology during the week of September 7--9, 2011. The conference brought together leading researchers and practitioners involved with all aspects of nonlinear mathematics for uncertainty and its applications. Over the last fifty years there have been many attempts in extending the theory of classical probability and statistical models to the generalized one which can cope with problems of inference and decision making when the model-related information is scarce, vague, ambiguous, or incomplete. Such attempts include the study of nonadditive measures and their integrals, imprecise probabilities and random sets, and their applications in information sciences, economics, finance, insurance, engineering, and social sciences. The book presents topics including nonadditive measures and nonlinear integrals, Choquet, Sugeno and other types of integrals, possibility theory, Dempster-Shafer theory, random sets, fuzzy random sets and related statistics, set-valued and fuzzy stochastic processes, imprecise probability theory and related statistical models, fuzzy mathematics, nonlinear functional analysis, information theory, mathematical finance and risk managements, decision making under various types of uncertainty, and others.
Categories: Computers

Functional Analysis

Functional Analysis

This is the fourth and final volume in the Princeton Lectures in Analysis, a series of textbooks that aim to present, in an integrated manner, the core areas of analysis.

Author: Elias M. Stein

Publisher: Princeton University Press

ISBN: 9781400840557

Category: Mathematics

Page: 448

View: 241

This is the fourth and final volume in the Princeton Lectures in Analysis, a series of textbooks that aim to present, in an integrated manner, the core areas of analysis. Beginning with the basic facts of functional analysis, this volume looks at Banach spaces, Lp spaces, and distribution theory, and highlights their roles in harmonic analysis. The authors then use the Baire category theorem to illustrate several points, including the existence of Besicovitch sets. The second half of the book introduces readers to other central topics in analysis, such as probability theory and Brownian motion, which culminates in the solution of Dirichlet's problem. The concluding chapters explore several complex variables and oscillatory integrals in Fourier analysis, and illustrate applications to such diverse areas as nonlinear dispersion equations and the problem of counting lattice points. Throughout the book, the authors focus on key results in each area and stress the organic unity of the subject. A comprehensive and authoritative text that treats some of the main topics of modern analysis A look at basic functional analysis and its applications in harmonic analysis, probability theory, and several complex variables Key results in each area discussed in relation to other areas of mathematics Highlights the organic unity of large areas of analysis traditionally split into subfields Interesting exercises and problems illustrate ideas Clear proofs provided
Categories: Mathematics

Notices of the American Mathematical Society

Notices of the American Mathematical Society

Hamdy, Hosny, Sequential estimation of the parameters of negative exponetial
and rectangular distributions. Moen ... Lehigh University (2;2,0,0,0,0,0,0)
MATHEMATICS Frantz, Deborah A., methods, probability distributions,
associated positive linear operators. ... Bator, Elizabeth Mary, Duals of separable
Banach spaces. ... )eisher, Caroline I., A survey of the applications of Pólya's
enumeration theorem.

Author: American Mathematical Society

Publisher:

ISBN: UCAL:B3647861

Category: Mathematics

Page:

View: 165

Categories: Mathematics

Introduction to Modern Analysis

Introduction to Modern Analysis

This text is based on lectures given by the author at the advanced undergraduate and graduate levels in Measure Theory, Functional Analysis, Banach Algebras, Spectral Theory (of bounded and unbounded operators), Semigroups of Operators, ...

Author: Shmuel Kantorovitz

Publisher: OUP Oxford

ISBN: 0191523550

Category: Mathematics

Page: 448

View: 570

This text is based on lectures given by the author at the advanced undergraduate and graduate levels in Measure Theory, Functional Analysis, Banach Algebras, Spectral Theory (of bounded and unbounded operators), Semigroups of Operators, Probability and Mathematical Statistics, and Partial Differential Equations. The first 10 chapters discuss theoretical methods in Measure Theory and Functional Analysis, and contain over 120 end of chapter exercises. The final two chapters apply theory to applications in Probability Theory and Partial Differential Equations. The Measure Theory chapters discuss the Lebesgue-Radon-Nikodym theorem which is given the Von Neumann Hilbert space proof. Also included are the relatively advanced topics of Haar measure, differentiability of complex Borel measures in Euclidean space with respect to Lebesgue measure, and the Marcinkiewicz' interpolation theorem for operators between Lebesgue spaces. The Functional Analysis chapters cover the usual material on Banach spaces, weak topologies, separation, extremal points, the Stone-Weierstrass theorem, Hilbert spaces, Banach algebras, and Spectral Theory for both bounded and unbounded operators. Relatively advanced topics such as the Gelfand-Naimark-Segal representation theorem and the Von Neumann double commutant theorem are included. The final two chapters deal with applications, where the measure theory and functional analysis methods of the first ten chapters are applied to Probability Theory and the Theory of Distributions and PDE's. Again, some advanced topics are included, such as the Lyapounov Central Limit theorem, the Kolmogoroff "Three Series theorem", the Ehrenpreis-Malgrange-Hormander theorem on fundamental solutions, and Hormander's theory of convolution operators. The Oxford Graduate Texts in Mathematics series aim is to publish textbooks suitable for graduate students in mathematics and its applications. The level of books may range from some suitable for advanced undergraduate courses at one end, to others of interest to research workers. The emphasis is on texts of high mathematical quality in active areas, particularly areas that are not well represented in the literature at present.
Categories: Mathematics