Probability Measures on Locally Compact Groups

Chapter III Embedding of Infinitely Divisible Probability Measures One of the most prominent subclasses of probability measures on a locally compact group is the class of infinitely divisible probability measures.

Author: H. Heyer

Publisher: Springer Science & Business Media

ISBN: 3642667066

Category: Mathematics

Page: 532

View: 213

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.

Probability 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.

Author: H. Heyer

Publisher: Springer

ISBN: 9783642667084

Category: Mathematics

Page: 532

View: 132

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.

Stable Probability Measures on Euclidean Spaces and on Locally Compact Groups

pact group contracting modulo a compact subgroup and applications to stable convolution semigroups. Semigroup Forum 33, 111–143 (1986) ... [153] Heyer, H.: Probability Measures on Locally Compact Groups. BerlinHeidelberg–New York.

Author: Wilfried Hazod

Publisher: Springer Science & Business Media

ISBN: 940173061X

Category: Mathematics

Page: 612

View: 503

Generalising classical concepts of probability theory, the investigation of operator (semi)-stable laws as possible limit distributions of operator-normalized sums of i.i.d. random variable on finite-dimensional vector space started in 1969. Currently, this theory is still in progress and promises interesting applications. Parallel to this, similar stability concepts for probabilities on groups were developed during recent decades. It turns out that the existence of suitable limit distributions has a strong impact on the structure of both the normalizing automorphisms and the underlying group. Indeed, investigations in limit laws led to contractable groups and - at least within the class of connected groups - to homogeneous groups, in particular to groups that are topologically isomorphic to a vector space. Moreover, it has been shown that (semi)-stable measures on groups have a vector space counterpart and vice versa. The purpose of this book is to describe the structure of limit laws and the limit behaviour of normalized i.i.d. random variables on groups and on finite-dimensional vector spaces from a common point of view. This will also shed a new light on the classical situation. Chapter 1 provides an introduction to stability problems on vector spaces. Chapter II is concerned with parallel investigations for homogeneous groups and in Chapter III the situation beyond homogeneous Lie groups is treated. Throughout, emphasis is laid on the description of features common to the group- and vector space situation. Chapter I can be understood by graduate students with some background knowledge in infinite divisibility. Readers of Chapters II and III are assumed to be familiar with basic techniques from probability theory on locally compact groups.

Probability Measures on Groups VIII

ERGODIC AND MIXING PROPERTIES OF MEASURES ON LOCALLY COMPACT GROUPS Eberhard Kani uth The ergodic and mixing properties of (probability) measures on locally compact abelian groups were characterized by Choquet and Deny [1] and by Foguel ...

Author: Herbert Heyer

Publisher: Springer

ISBN: 3540448527

Category: Mathematics

Page: 388

View: 308


Probability Measures on Semigroups Convolution Products Random Walks and Random Matrices

Collins, H. S., “Idempotent measures on compact semigroups,” Proc. Amer Math. Soc. 13,442—446 (1962). Csiszar, I., “On infinite products of random elements and infinite convolutions of probability distributions on locally compact groups ...

Author: Göran Högnäs

Publisher: Springer Science & Business Media

ISBN: 1475723881

Category: Mathematics

Page: 388

View: 725

A Scientific American article on chaos, see Crutchfield et al. (1986), illus trates a very persuasive example of recurrence. A painting of Henri Poincare, or rather a digitized version of it, is stretched and cut to produce a mildly distorted image of Poincare. The same procedure is applied to the distorted image and the process is repeated over and over again on the successively more and more blurred images. After a dozen repetitions nothing seems to be left of the original portrait. Miraculously, structured images appear briefly as we continue to apply the distortion procedure to successive images. After 241 iterations the original picture reappears, unchanged! Apparently the pixels of the Poincare portrait were moving about in accor dance with a strictly deterministic rule. More importantly, the set of all pixels, the whole portrait, was transformed by the distortion mechanism. In this exam ple the transformation seems to have been a reversible one since the original was faithfully recreated. It is not very farfetched to introduce a certain amount of randomness and irreversibility in the above example. Think of a random miscoloring of some pixels or of inadvertently giving a pixel the color of its neighbor. The methods in this book are geared towards being applicable to the asymp totics of such transformation processes. The transformations form a semigroup in a natural way; we want to investigate the long-term behavior of random elements of this semigroup.

Stable Probability Measures on Euclidean Spaces and on Locally Compact Groups

The purpose of this book is to describe the structure of limit laws and the limit behaviour of normalized i.i.d. random variables on groups and on finite-dimensional vector spaces from a common point of view.

Author: Wilfried Hazod

Publisher: Springer Science & Business Media

ISBN: 9781402000409

Category: Mathematics

Page: 668

View: 360

Generalising classical concepts of probability theory, the investigation of operator (semi)-stable laws as possible limit distributions of operator-normalized sums of i.i.d. random variable on finite-dimensional vector space started in 1969. Currently, this theory is still in progress and promises interesting applications. Parallel to this, similar stability concepts for probabilities on groups were developed during recent decades. It turns out that the existence of suitable limit distributions has a strong impact on the structure of both the normalizing automorphisms and the underlying group. Indeed, investigations in limit laws led to contractable groups and - at least within the class of connected groups - to homogeneous groups, in particular to groups that are topologically isomorphic to a vector space. Moreover, it has been shown that (semi)-stable measures on groups have a vector space counterpart and vice versa. The purpose of this book is to describe the structure of limit laws and the limit behaviour of normalized i.i.d. random variables on groups and on finite-dimensional vector spaces from a common point of view. This will also shed a new light on the classical situation. Chapter 1 provides an introduction to stability problems on vector spaces. Chapter II is concerned with parallel investigations for homogeneous groups and in Chapter III the situation beyond homogeneous Lie groups is treated. Throughout, emphasis is laid on the description of features common to the group- and vector space situation. Chapter I can be understood by graduate students with some background knowledge in infinite divisibility. Readers of Chapters II and III are assumed to be familiar with basic techniques from probability theory on locally compact groups.

Harmonic Analysis of Probability Measures on Hypergroups

Heyer. Herbert. Probability measures on locally compact groups. Ergebnisse der Mathematik und ihrer Grenzgebiete. 94. Springer, Berlin.-bloat York. 1911. Heyer. Herbert. iltioments oy' probability measures on a group. lntemat. J. Math.

Author: Walter R. Bloom

Publisher: Walter de Gruyter

ISBN: 3110877597

Category: Mathematics

Page: 607

View: 870

The series is devoted to the publication of monographs and high-level textbooks in mathematics, mathematical methods and their applications. Apart from covering important areas of current interest, a major aim is to make topics of an interdisciplinary nature accessible to the non-specialist. The works in this series are addressed to advanced students and researchers in mathematics and theoretical physics. In addition, it can serve as a guide for lectures and seminars on a graduate level. The series de Gruyter Studies in Mathematics was founded ca. 30 years ago by the late Professor Heinz Bauer and Professor Peter Gabriel with the aim to establish a series of monographs and textbooks of high standard, written by scholars with an international reputation presenting current fields of research in pure and applied mathematics. While the editorial board of the Studies has changed with the years, the aspirations of the Studies are unchanged. In times of rapid growth of mathematical knowledge carefully written monographs and textbooks written by experts are needed more than ever, not least to pave the way for the next generation of mathematicians. In this sense the editorial board and the publisher of the Studies are devoted to continue the Studies as a service to the mathematical community. Please submit any book proposals to Niels Jacob.

Probability Measures on Groups X

12. 13. 14. Hazo d, W. , Stable probabilities on locally compact groups. In Probability Measures on Groups. Proceedings Oberwolfach 1981. Lecture Notes Math. 928, 183 – 211 (1983). Hazo d, W. , Remarks on [semi- ) stable probabilities.

Author: H. Heyer

Publisher: Springer Science & Business Media

ISBN: 1489923640

Category: Mathematics

Page: 498

View: 868

The present volume contains the transactions of the lOth Oberwolfach Conference on "Probability Measures on Groups". The series of these meetings inaugurated in 1970 by L. Schmetterer and the editor is devoted to an intensive exchange of ideas on a subject which developed from the relations between various topics of mathematics: measure theory, probability theory, group theory, harmonic analysis, special functions, partial differential operators, quantum stochastics, just to name the most significant ones. Over the years the fruitful interplay broadened in various directions: new group-related structures such as convolution algebras, generalized translation spaces, hypercomplex systems, and hypergroups arose from generalizations as well as from applications, and a gradual refinement of the combinatorial, Banach-algebraic and Fourier analytic methods led to more precise insights into the theory. In a period of highest specialization in scientific thought the separated minds should be reunited by actively emphasizing similarities, analogies and coincidences between ideas in their fields of research. Although there is no real separation between one field and another - David Hilbert denied even the existence of any difference between pure and applied mathematics - bridges between probability theory on one side and algebra, topology and geometry on the other side remain absolutely necessary. They provide a favorable ground for the communication between apparently disjoint research groups and motivate the framework of what is nowadays called "Structural probability theory".

Probability Measures on Groups

79, 25–1, 5 (1975) HEYER H. : "Probability measures on locally compact groups : Ergebnisse der Math. Berlin-Heidelberg-New York, Springer 1977 JANSSEN A. : "Charakterisierung stetiger Faltungshalbgruppen durch das Lévy-Maso". Math.

Author: H. Heyer

Publisher: Springer

ISBN: 3540392068

Category: Mathematics

Page: 480

View: 550

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Probability Measures on Metric Spaces

PRELIMINARY FACTS ABOUT A GROUP AND ITS CHARACTER GROUP Let X be a locally compact second countable abelian group and Y its character group. Y consists of all continuous homomorphisms from X into the circle group in the complex plane.

Author: K. R. Parthasarathy

Publisher: Academic Press

ISBN: 1483225259

Category: Mathematics

Page: 288

View: 305

Probability Measures on Metric Spaces presents the general theory of probability measures in abstract metric spaces. This book deals with complete separable metric groups, locally impact abelian groups, Hilbert spaces, and the spaces of continuous functions. Organized into seven chapters, this book begins with an overview of isomorphism theorem, which states that two Borel subsets of complete separable metric spaces are isomorphic if and only if they have the same cardinality. This text then deals with properties such as tightness, regularity, and perfectness of measures defined on metric spaces. Other chapters consider the arithmetic of probability distributions in topological groups. This book discusses as well the proofs of the classical extension theorems and existence of conditional and regular conditional probabilities in standard Borel spaces. The final chapter deals with the compactness criteria for sets of probability measures and their applications to testing statistical hypotheses. This book is a valuable resource for statisticians.