Probability Measures on Locally Compact Groups

ISBN-13: 9783642667084 / Angielski / Miękka / 2011 / 532 str.

H. Heyer

Opis

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."

Spis treści

Preliminaries. Almost Periodic Locally Compact Groups.- References and Comments.- I. Harmonic Analysis of Almost Periodic Locally Compact Groups.- 1.1 Measures on a Locally Compact Space.- 1.2 Convolution of Measures on a Locally Compact Group.- 1.3 Fourier Transforms of Bounded Measures.- 1.4 The Theorems of Lévy and Bochner.- 1.5 Convolution Semigroups and Negative-Definite Forms.- References and Comments.- II. Convergence of Convolution Sequences of Probability Measures.- 2.1 Convolution Powers on a Compact Group.- 2.2 Equivalence of Types of Convergence.- 2.3 The Normed Convergence Property.- 2.4 Convergence in Variance.- 2.5 Asymptotic Equidistribution.- 2.6 Shifting Iterated Convolutions.- References and Comments.- III. Embedding of Infinitely Divisible Probability Measures.- 3.1 Root Compact Groups.- 3.2 Poisson Measures and Their Characterizations.- 3.3 Submonogeneous Embedding of Infinitely Divisible Measures.- 3.4 Existence of One-Parameter Semigroups.- 3.5 The General Continuous Embedding.- 3.6 Injective Submonogeneous Embeddings.- References and Comments.- IV. Canonical Representations of Convolution Semigroups.- 4.1 Positive Semigroups and Their Generating Functionals.- 4.2 Hunt’s Representation Theorem.- 4.3 The Lévy-Khintchine Formula for Almost Periodic Groups.- 4.4 The Canonical Representation of Almost Positive Functionals.- 4.5 The Lévy-Khintchine Formula for General Locally Compact Groups.- 4.6 Convolution Hemigroups. Generation and Representation.- References and Comments.- V. The Central Limit Problem in the Abelian Case.- 5.1 Convergence of Infinitesimal Systems.- 5.2 Gauss Measures in the Sense of Parthasarathy.- 5.3 Gauss Measures in the Sense of Bernstein.- 5.4 Convergence to Gauss Measures.- 5.5 Symmetric Gauss Semigroups.- 5.6 Additive Processes and Their Decomposition.- References and Comments.- VI. The Central Limit Problem in the General Case.- 6.1 Poisson Embedding and Approximation.- 6.2 Gauss Measures and Their Characterizations.- 6.3 Absolute Continuity and Diffusion of Gauss Semigroups.- 6.4 Central Gauss Semigroups.- 6.5 Convergence of Triangular Systems of Probability Measures.- 6.6 Central Limit Theorems for Totally Disconnected Groups.- References and Comments.- List of Symbols.