Probabilistic Analysis and Related Topics

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.

List of Contributors

Preface

Optimal Control of Stochastic Systems

I. Introduction

II. Existence of Optimal Controls without Feedback

III. Existence of Optimal Feedback Controls

IV. Random Necessary Conditions

V. Analytic Necessary Conditions

References

Gleason Measures

I. Introduction

II. Generalities

III. Orthogonally Scattered Gleason Measures

IV. L2ξ(H)-Spaces and Isometries Generated by OSG Measures

V. Spectral Gleason Measures

VI. Convergence of Gleason Measures

VII. Gleason Measures in Tensor Products

VIII. Random Gleason Measures

References

An Introduction to Nonstandard Analysis and Hyperfinite Probability Theory

I. Introduction

II. An Introduction to Nonstandard Analysis

III. A Nonstandard Representation of Measurable Spaces and L∞

IV. Conversion from Nonstandard to Standard Measure Spaces

V. Applications to Stochastic Processes

References

Limit Theorems: Stochastic Matrices, Ergodic Markov Chains, and Measures on Semigroups

I. Introduction and Preliminaries

II. Limits of Convolutions in Groups and Semigroups: Analysis in Stochastic Matrices

III. Ergodicity of Markov Chains and Probability Measures on Semigroups: An Interplay

IV. Limit Theorems for Convolution Products of Probability Measures on Completely Simple Semigroups

References

Index