Set-Indexed Martingales

Set-Indexed Martingales

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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.
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Product details

  • Hardback | 224 pages
  • 158 x 234 x 18mm | 480.82g
  • Taylor & Francis Inc
  • Chapman & Hall/CRC
  • United States
  • English
  • 1584880821
  • 9781584880820

Table of contents

Introduction General Theory Generalities. Predictability. Martingales. Decompositions and Quadratic Variation Martingale Characterizations. Generalizations of Martingales Weak Convergence. Weak Convergence of Set-Indexed Processes Limit Theorems for Point Processes Martingale Central Limit Theorems References Index.
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Review quote

"a small, elegant volumeThis 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
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