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.
"…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
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