Characterization Of Probability Distributions On Locally Compact Abelian Groups

Author : Gennadiy Feldman
Publisher : American Mathematical Society
Page : 253 pages
File Size : 51,5 Mb
Release : 2023-04-07
Category : Mathematics
ISBN : 9781470472955

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Characterization of Probability Distributions on Locally Compact Abelian Groups by Gennadiy Feldman Pdf

It is well known that if two independent identically distributed random variables are Gaussian, then their sum and difference are also independent. It turns out that only Gaussian random variables have such property. This statement, known as the famous Kac-Bernstein theorem, is a typical example of a so-called characterization theorem. Characterization theorems in mathematical statistics are statements in which the description of possible distributions of random variables follows from properties of some functions of these random variables. The first results in this area are associated with famous 20th century mathematicians such as G. Pólya, M. Kac, S. N. Bernstein, and Yu. V. Linnik. By now, the corresponding theory on the real line has basically been constructed. The problem of extending the classical characterization theorems to various algebraic structures has been actively studied in recent decades. The purpose of this book is to provide a comprehensive and self-contained overview of the current state of the theory of characterization problems on locally compact Abelian groups. The book will be useful to everyone with some familiarity of abstract harmonic analysis who is interested in probability distributions and functional equations on groups.

Author : Gennadiĭ Mikhaĭlovich Felʹdman
Publisher : Unknown
Page : 0 pages
File Size : 50,9 Mb
Release : 2023
Category : Distribution (Probability theory)
ISBN : 1470473267

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Characterization of Probability Distributions on Locally Compact Abelian Groups by Gennadiĭ Mikhaĭlovich Felʹdman Pdf

Author : Gennadiĭ Mikhaĭlovich Felʹdman
Publisher : European Mathematical Society
Page : 272 pages
File Size : 41,6 Mb
Release : 2008
Category : Abelian groups
ISBN : 3037190450

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Functional Equations and Characterization Problems on Locally Compact Abelian Groups by Gennadiĭ Mikhaĭlovich Felʹdman Pdf

This book deals with the characterization of probability distributions. It is well known that both the sum and the difference of two Gaussian independent random variables with equal variance are independent as well. The converse statement was proved independently by M. Kac and S. N. Bernstein. This result is a famous example of a characterization theorem. In general, characterization problems in mathematical statistics are statements in which the description of possible distributions of random variables follows from properties of some functions in these variables. In recent years, a great deal of attention has been focused upon generalizing the classical characterization theorems to random variables with values in various algebraic structures such as locally compact Abelian groups, Lie groups, quantum groups, or symmetric spaces. The present book is aimed at the generalization of some well-known characterization theorems to the case of independent random variables taking values in a locally compact Abelian group $X$. The main attention is paid to the characterization of the Gaussian and the idempotent distribution (group analogs of the Kac-Bernstein, Skitovich-Darmois, and Heyde theorems). The solution of the corresponding problems is reduced to the solution of some functional equations in the class of continuous positive definite functions defined on the character group of $X$. Group analogs of the Cramer and Marcinkiewicz theorems are also studied. The author is an expert in algebraic probability theory. His comprehensive and self-contained monograph is addressed to mathematicians working in probability theory on algebraic structures, abstract harmonic analysis, and functional equations. The book concludes with comments and unsolved problems that provide further stimulation for future research in the theory.

Author : Gennadiĭ Mikhaĭlovich Felʹdman
Publisher : American Mathematical Soc.
Page : 236 pages
File Size : 44,5 Mb
Release : 1993
Category : Mathematics
ISBN : 0821845934

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Arithmetic of Probability Distributions, and Characterization Problems on Abelian Groups by Gennadiĭ Mikhaĭlovich Felʹdman Pdf

This book studies the problem of the decomposition of a given random variable introduction a sum of independent random variables (components). The central feature of the book is Feldman's use of powerful analytical techniques.

Author : Gennadij M. Fel'dman
Publisher : American Mathematical Soc.
Page : 236 pages
File Size : 47,7 Mb
Release : 2024-07-04
Category : Mathematics
ISBN : 0821897446

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Arithmetic of Probability Distributions, and Characterization Problems on Abelian Groups by Gennadij M. Fel'dman Pdf

This book studies the problem of the decomposition of a given random variable introduction a sum of independent random variables (components). The central feature of the book is Feldman's use of powerful analytical techniques.

Author : H. Heyer
Publisher : Springer Science & Business Media
Page : 542 pages
File Size : 41,8 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9783642667060

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Probability Measures on Locally Compact Groups by H. Heyer Pdf

Probability measures on algebraic-topological structures such as topological semi groups, groups, and vector spaces have become of increasing importance in recent years for probabilists interested in the structural aspects of the theory as well as for analysts aiming at applications within the scope of probability theory. In order to obtain a natural framework for a first systematic presentation of the most developed part of the work done in the field we restrict ourselves to prob ability measures on locally compact groups. At the same time we stress the non Abelian aspect. Thus the book is concerned with a set of problems which can be regarded either from the probabilistic or from the harmonic-analytic point of view. In fact, it seems to be the synthesis of these two viewpoints, the initial inspiration coming from probability and the refined techniques from harmonic analysis which made this newly established subject so fascinating. The goal of the presentation is to give a fairly complete treatment of the central limit problem for probability measures on a locally compact group. In analogy to the classical theory the discussion is centered around the infinitely divisible probability measures on the group and their relationship to the convergence of infinitesimal triangular systems.

Author : Ganapati P. Patil,S. Kotz,J.K. Ord
Publisher : Springer Science & Business Media
Page : 430 pages
File Size : 49,7 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9789401018487

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A Modern Course on Statistical Distributions in Scientific Work by Ganapati P. Patil,S. Kotz,J.K. Ord Pdf

These three volumes constitute the edited Proceedings of the NATO Advanced Study Institute on Statistical Distributions in Scientific Work held at the University of Calgary from July 29 to August 10, ~. 974. The general title of the volumes is "Statistical Distributions in Scientific Work". The individual volumes are: Volume 1 - Models and Structures; Volume 2 - Model Building and Model Selection; and Volume 3 - Characterizations and Applications. These correspond to the three advanced seminars of the Institute devoted to the respective subject areas. The planned activities of the Institute consisted of main lectures and expositions, seminar lectures and study group dis cussions, tutorials and individual study. The activities included meetings of editorial committees to discuss editorial matters for these proceedings which consist of contributions that have gone through the usual refereeing process. A special session was organized to consider the potential of introducing a course on statistical distributions in scientific modeling in the curriculum of statistics and quantitative studies. This session is reported in Volume 2. The overall perspective for the Institute is provided by the Institute Director, Professor G. P. Pati1, in his inaugural address which appears in Volume 1. The Linnik Memorial Inaugural Lecture given by Professor C. R. Rao for the Characterizations Seminar is included in Volume 3. As discussed in the Institute inaugural address, not mL.

Author : Bozzano G Luisa
Publisher : Academic Press
Page : 271 pages
File Size : 54,8 Mb
Release : 2012-09-18
Category : Mathematics
ISBN : 9780128015261

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Identifiability In Stochastic Models by Bozzano G Luisa Pdf

The problem of identifiability is basic to all statistical methods and data analysis, occurring in such diverse areas as Reliability Theory, Survival Analysis, and Econometrics, where stochastic modeling is widely used. Mathematics dealing with identifiability per se is closely related to the so-called branch of "characterization problems" in Probability Theory. This book brings together relevant material on identifiability as it occurs in these diverse fields.

Author : Herbert Heyer
Publisher : World Scientific
Page : 425 pages
File Size : 41,5 Mb
Release : 2009
Category : Mathematics
ISBN : 9789814282499

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Structural Aspects in the Theory of Probability by Herbert Heyer Pdf

The book is conceived as a text accompanying the traditional graduate courses on probability theory. An important feature of this enlarged version is the emphasis on algebraic-topological aspects leading to a wider and deeper understanding of basic theorems such as those on the structure of continuous convolution semigroups and the corresponding processes with independent increments. Fourier transformation OCo the method applied within the settings of Banach spaces, locally compact Abelian groups and commutative hypergroups OCo is given an in-depth discussion. This powerful analytic tool along with the relevant facts of harmonic analysis make it possible to study certain properties of stochastic processes in dependence of the algebraic-topological structure of their state spaces. In extension of the first edition, the new edition contains chapters on the probability theory of generalized convolution structures such as polynomial and Sturm-Liouville hypergroups, and on the central limit problem for groups such as tori, p-adic groups and solenoids. Sample Chapter(s). Chapter 1: Probability Measures on Metric Spaces (318 KB). Contents: Probability Measures on Metric Spaces; The Fourier Transform in a Banach Space; The Structure of Infinitely Divisible Probability Measures; Harmonic Analysis of Convolution Semigroups; Negative Definite Functions and Convolution Semigroups; Probabilistic Properties of Convolution Semigroups; Hypergroups in Probability Theory; Limit Theorems on Locally Compact Abelian Groups. Readership: Graduate students, lecturers and researchers in probability and statistics."

Author : Martin Engert
Publisher : Unknown
Page : 134 pages
File Size : 53,8 Mb
Release : 1965
Category : Abelian groups
ISBN : STANFORD:36105025548285

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A Characterization of Differential Operators on Locally Compact Abelian Groups by Martin Engert Pdf

Author : Inna Capdeboscq,Daniel Gorenstein,Richard Lyons,Ronald Solomon
Publisher : American Mathematical Society
Page : 587 pages
File Size : 52,9 Mb
Release : 2023-10-23
Category : Mathematics
ISBN : 9781470475536

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The Classification of the Finite Simple Groups, Number 10 by Inna Capdeboscq,Daniel Gorenstein,Richard Lyons,Ronald Solomon Pdf

This book is the tenth in a series of volumes whose aim is to provide a complete proof of the classification theorem for the finite simple groups based on a fairly short and clearly enumerated set of background results. Specifically, this book completes our identification of the simple groups of bicharacteristic type begun in the ninth volume of the series (see SURV/40.9). This is a fascinating set of simple groups which have properties in common with matrix groups (or, more generally, groups of Lie type) defined both over fields of characteristic 2 and over fields of characteristic 3. This set includes 11 of the celebrated 26 sporadic simple groups along with several of their large simple subgroups. Together with SURV/40.9, this volume provides the first unified treatment of this class of simple groups.

Author : Alekos Vidras,Alain Yger
Publisher : American Mathematical Society
Page : 556 pages
File Size : 51,5 Mb
Release : 2023-10-18
Category : Mathematics
ISBN : 9781470471125

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Multidimensional Residue Theory and Applications by Alekos Vidras,Alain Yger Pdf

Residue theory is an active area of complex analysis with connections and applications to fields as diverse as partial differential and integral equations, computer algebra, arithmetic or diophantine geometry, and mathematical physics. Multidimensional Residue Theory and Applications defines and studies multidimensional residues via analytic continuation for holomorphic bundle-valued current maps. This point of view offers versatility and flexibility to the tools and constructions proposed, allowing these residues to be defined and studied outside the classical case of complete intersection. The book goes on to show how these residues are algebraic in nature, and how they relate and apply to a wide range of situations, most notably to membership problems, such as the Briançon–Skoda theorem and Hilbert's Nullstellensatz, to arithmetic intersection theory and to tropical geometry. This book will supersede the existing literature in this area, which dates back more than three decades. It will be appreciated by mathematicians and graduate students in multivariate complex analysis. But thanks to the gentle treatment of the one-dimensional case in Chapter 1 and the rich background material in the appendices, it may also be read by specialists in arithmetic, diophantine, or tropical geometry, as well as in mathematical physics or computer algebra.

Author : A. T. Bharucha-Reid
Publisher : Elsevier
Page : 220 pages
File Size : 41,9 Mb
Release : 2014-05-10
Category : Mathematics
ISBN : 9781483275536

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Probabilistic Analysis and Related Topics by A. T. Bharucha-Reid Pdf

Probabilistic Analysis and Related Topics, Volume 2 focuses on the integrability, continuity, and differentiability of random functions, as well as functional analysis, measure theory, operator theory, and numerical analysis. The selection first offers information on the optimal control of stochastic systems and Gleason measures. Discussions focus on convergence of Gleason measures, random Gleason measures, orthogonally scattered Gleason measures, existence of optimal controls without feedback, random necessary conditions, and Gleason measures in tensor products. The text then elaborates on an introduction to nonstandard analysis and hyperfinite probability theory, including applications to stochastic processes, conversion from nonstandard to standard measure spaces, and an introduction to nonstandard analysis. The text examines stochastic matrices, ergodic Markov chains, and measures on semigroups, as well as limit theorems for convolution products of probability measures on completely simple semigroups; ergodicity of Markov chains and probability measures on semigroups; and limits of convolutions in groups and semigroups. The selection is a dependable source of data for mathematicians and researchers interested in the general theory of random functions.

Author : Ellen Elizabeth Eischen,Wee Teck Gan,Aaron Pollack,Zhiwei Yun
Publisher : American Mathematical Society
Page : 199 pages
File Size : 50,8 Mb
Release : 2024-03-26
Category : Mathematics
ISBN : 9781470474928

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Automorphic Forms Beyond $mathrm {GL}_2$ by Ellen Elizabeth Eischen,Wee Teck Gan,Aaron Pollack,Zhiwei Yun Pdf

The Langlands program has been a very active and central field in mathematics ever since its conception over 50 years ago. It connects number theory, representation theory and arithmetic geometry, and other fields in a profound way. There are nevertheless very few expository accounts beyond the GL(2) case. This book features expository accounts of several topics on automorphic forms on higher rank groups, including rationality questions on unitary group, theta lifts and their applications to Arthur's conjectures, quaternionic modular forms, and automorphic forms over functions fields and their applications to inverse Galois problems. It is based on the lecture notes prepared for the twenty-fifth Arizona Winter School on “Automorphic Forms beyond GL(2)”, held March 5–9, 2022, at the University of Arizona in Tucson. The speakers were Ellen Eischen, Wee Teck Gan, Aaron Pollack, and Zhiwei Yun. The exposition of the book is in a style accessible to students entering the field. Advanced graduate students as well as researchers will find this a valuable introduction to various important and very active research areas.

Author : Balázs Bárány,Károly Simon,Boris Solomyak
Publisher : American Mathematical Society
Page : 466 pages
File Size : 47,7 Mb
Release : 2023-11-16
Category : Mathematics
ISBN : 9781470470463

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Self-similar and Self-affine Sets and Measures by Balázs Bárány,Károly Simon,Boris Solomyak Pdf

Although there is no precise definition of a “fractal”, it is usually understood to be a set whose smaller parts, when magnified, resemble the whole. Self-similar and self-affine sets are those for which this resemblance is precise and given by a contracting similitude or affine transformation. The present book is devoted to this most basic class of fractal objects. The book contains both introductory material for beginners and more advanced topics, which continue to be the focus of active research. Among the latter are self-similar sets and measures with overlaps, including the much-studied infinite Bernoulli convolutions. Self-affine systems pose additional challenges; their study is often based on ergodic theory and dynamical systems methods. In the last twenty years there have been many breakthroughs in these fields, and our aim is to give introduction to some of them, often in the simplest nontrivial cases. The book is intended for a wide audience of mathematicians interested in fractal geometry, including students. Parts of the book can be used for graduate and even advanced undergraduate courses.