Set-Indexed Martingales

Gail Ivanoff, Ely Merzbach, B.G. Ivanoff

• Publisher: CRC Press
• Year: 2024
• eBook ISBN: 9781351416801
• Edition: 0
• Topic: Mathematics
• Subtopic: Probability & Statistics
• Index: Mathematics

Set-Indexed Martingales offers a unique, comprehensive development of a general theory of Martingales indexed by a family of sets. The authors establish-for the first time-an appropriate framework that provides a suitable structure for a theory of Martingales with enough generality to include many interesting examples. Developed from first principles, the theory brings together the theories of Martingales with a directed index set and set-indexed stochastic processes. Part One presents several classical concepts extended to this setting, including: stopping, predictability, Doob-Meyer decompositions, martingale characterizations of the set-indexed Poisson process, and Brownian motion. Part Two addresses convergence of sequences of set-indexed processes and introduces functional convergence for processes whose sample paths live in a Skorokhod-type space and semi-functional convergence for processes whose sample paths may be badly behaved.Completely self-contained, the theoretical aspects of this work are rich and promising. With its many important applications-especially in the theory of spatial statistics and in stochastic geometry- Set Indexed Martingales will undoubtedly generate great interest and inspire further research and development of the theory and applications.

Table of contents

• Cover
• Title Page
• Copyright Page
• Dedication
• Table of Contents
• Acknowledgements
• Introduction
• I General Theory
• II Weak Convergence
• References
• Index