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
Generalities. Predictability. Martingales. Decompositions and Quadratic Variation
Martingale Characterizations. Generalizations of Martingales
Weak Convergence of Set-Indexed Processes
Limit Theorems for Point Processes
Martingale Central Limit Theorems
Ivanoff, B.G.; Merzbach, Ely
"…a small, elegant volume…This state-of-the-art monograph will be a valuable resource and stimulus for further work in the area."
-Short Book Reviews of the ISI
"I would recommend the book as an excellent introduction to set-indexed martingales. The foundations of the general theory are clearly presented and the reader is led to a point that is close to the current edge of research."
--Simon Harris, University of Bath