Stable probability measures on Euclidean spaces and on locally compact groups : structural properties and limit theorems

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Stable probability measures on Euclidean spaces and on locally compact groups : structural properties and limit theorems / by Wilfried Hazod and Eberhard Siebert

資料種別:

図書

出版情報:

Dordrecht : Kluwer Academic Publishers, c2001

形態:

xvii, 612 p. ; 25 cm

シリーズ名:

Mathematics and its applications ; vol. 531 <BA00044495>

著者名:

ISBN:

9781402000409 [1402000405]

書誌ID:

BA5412787X

子書誌情報

フルテキスト

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他の版・巻

書誌詳細

注記:

Includes bibliographical references (p. 573-599) and index

言語:

英語

目次情報:

Preface

Introduction

Probabilities on vector spaces / I：

Preparations: Linear operators on finite-dimensional vector spaces / 1.1：

Notations (in particular for Chapter I)

Discrete one-parameter groups of operators / II：

Continuous one-parameter groups of operators / III：

Linear groups / IV：

Full probability measures and convergence of types / 1.2：

Operator-semistable laws and operator-stable laws / 1.3：

Definition and Levy-Khinchin representation

Annexe: More on infinitely divisible laws

Levy measures of operator- (semi-) stable laws / 1.4：

Levy measures of operator-semistable laws

Levy measures of operator-stable laws

Algebraic characterization of operator- (semi-) stability / 1.5：

The structure of Lin ([mu])

Subordination and (semi-) stability

A randomized characterization of operator-stability

Operator- (semi-) stable laws as limit distributions / 1.6：

Domains of operator- (semi-) attraction

Annexe: More on limits of infinitely divisible laws

More on domains of operator semi-attraction

Properties of operator- (semi-) stable laws / 1.7：

Exponents of operator-stable laws / 1.8：

Elliptical symmetry and large symmetry groups / 1.9：

Elliptically symmetric operator- (semi-) stable laws

Large symmetry groups

Domains of normal operator attraction / 1.10：

Stable laws

Remarks on operator-semistable laws

Moments and domains of attraction

The existence of commuting normalizations / 1.11：

More on the structure of the decomposability group Lin([mu]) / 1.12：

Semistability and strict semistability

Jordan decomposition and spectrum of normalizing operators

Marginal distributions of operator (semi-) stable laws

More on convergence of types theorems / 1.13：

Types and transformation groups

Applications of the convergence of types theorem

Finite-dimensional vector spaces

A method to construct full measures, given B

Some examples / V：

Stochastic compactness and regular variation properties / VI：

Probabilities with idempotent type. [Gamma]-stable and completely stable measures / 1.14：

Examples and counterexamples / 1.15：

Operator-stable laws on V = R[superscript 2] and R[superscript 3]

Subordination of stable laws

Probabilities with discrete symmetry group on V = R[superscript 2]

Marginal distributions of operator stable laws

Convergence of types and idempotent types

Limit laws and domains of attraction

Commuting normalizations / VII：

References and comments for Chapter I / 1.16：

Probabilities on simply connected nilpotent Lie groups

Probabilities on locally compact groups: Some fundamental theorems / 2.0：

Continuous convolution semigroups and the structure of generating functionals

Convergence of continuous convolution semigroups

Discrete convolution semigroups

Embedding theorems

Annexe: Supports of convolution semigroups

Discrete and continuous convolution semigroups: The translation procedure / 2.1：

Automorphisms and contractible Lie groups. Some basic facts

Some examples of contractible Lie groups

Convergence of types and full measures / 2.2：

Simply connected nilpotent Lie groups

Some generalizations

Semistable and stable continuous convolution semigroups on simply connected nilpotent Lie groups / 2.3：

Levy measures of stable and semistable laws / 2.4：

Levy measures of semistable laws

Levy measures of stable laws

Algebraic characterization of (semi-) stability / 2.5：

The structure of Lin([mu])

The structure of Inv([mu]), resp. Inv(A)

Subordination and semistability

(Semi-) stable laws as limit distributions / 2.6：

Limit theorems and uniqueness of embedding for semistable laws

Domains of (semi-) attraction

Properties of (semi-) stable laws / 2.7：

Absolute continuity and purity laws

Gaussian and Bochner stable measures

Holomorphic convolution semigroups

Moments of (semi-) stable laws

Exponents of stable laws / 2.8：

Elliptical symmetry and large invariance groups / 2.9：

Domains of normal attraction / 2.10：

Stable and semistable laws

Probabilities with idempotent type: [Gamma]-stable and completely stable measures / 2.11：

Idempotent (infinitesimal) [Gamma]-types and [Gamma]-stable laws

Complete stability

Marginals and complete stability

Intrinsic definitions of semistability

Domains of partial attraction and random limit theorems on groups and vector spaces / 2.12：

The existence of universal laws (Doeblin laws)

Stochastic compactness

Random limit theorems: Independent random times

Geometric (semi-) stability / 2.13：

Geometric convolutions

Properties of geometric and exponential distributions

Characterization of geometric convolutions and exponential mixtures

Geometric semistability

Geometric domains of attraction

Illustrations and examples for vector spaces G = V

More arithmetic properties of geometric convolutions

Remarks on self-decomposable laws on vector spaces and on groups / 2.14：

The decomposability semigroup D([mu])

Self-decomposability

Cocycle equations, background driving processes and generalized Ornstein-Uhlenbeck processes

Stable hemigroups and self-similar processes

Space-time processes

Processes on G and on V

Background driving processes with logarithmic moments

Full self-decomposable distributions and limit laws / VIII：

Generalizations and examples / IX：

More limit theorems on G and V: Mixing properties and dependent random times / 2.15：

A theorem of H. Cramer

Limit theorems for mixing arrays of random variables

Random limit theorems in the domain of attraction of (semi-) stable laws: Dependent random times

References and comments for Chapter II / 2.16：

(Semi-) stability and limit theorems on general locally compact groups

Contractive automorphisms on locally compact groups / 3.1：

Contractive automorphisms and contractible groups

Totally disconnected contractible groups

The structure theorem for contractible groups

Contractive one-parameter automorphism groups

Some more structure theorems for discrete automorphism groups

Automorphisms contracting modulo a compact subgroup K / 3.2：

Contraction mod K

The structure theorem: C[subscript K]([tau]) = C([tau])[middle dot]K for discrete automorphism groups acting on a Lie group

Borel cross-sections for the action of C([tau]) on C[subscript K]([tau]) (discrete automorphism groups)

Continuous automorphism groups

The structure theorem: C[subscript K](T) = C(T) [times sign, right closed] K for continuous automorphism groups

The structure of C[subscript K]([tau]) for p-adic Lie groups

Examples, counterexamples and some more structure theory / 3.3：

Contractible and K-contractible Lie groups

Automorphisms of compact groups

Infinite-dimensional tori and solenoidal groups

Retopologization of C([tau]): Intrinsic topologies of contractible groups

(Semi-) stable convolution semigroups with trivial idempotents / 3.4：

General definitions of strictly (semi-) stable convolution semigroups

(Semi-) stable continuous convolution semigroups with trivial idempotents

Some examples and further remarks

(Semi-) stable convolution semigroups with nontrivial idempotents / 3.5：

Semistable convolution semigroups on Lie groups with nontrivial idempotents

Stable convolution semigroups with nontrivial idempotents

Semistable submonogeneous semigroups on Lie groups

Semistable convolution semigroups with nontrivial idempotents on p-adic Lie groups

More on probabilities on contractible groups / 3.6：

Domains of partial attraction on contractible groups

The existence of Doeblin laws on contractible groups

A translation procedure for contractible locally compact groups

Point processes on groups and continuous convolution semigroups

Limit laws and convergence of types theorems. A survey / 3.7：

Limits of discrete convolution semigroups with nontrivial idempotents

Convergence of types theorems

Applications to semistability

Limit laws on compact extensions of contractible groups N [times sign, right closed] K

References and comments for Chapter III / 3.8：

Epilogue

Bibliography

List of Symbols

Index

Preface

Introduction

Probabilities on vector spaces / I：

Preparations: Linear operators on finite-dimensional vector spaces / 1.1：

Notations (in particular for Chapter I)

Discrete one-parameter groups of operators / II：

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