Set-Indexed Martingales: 1st Edition (Hardback) book cover

Set-Indexed Martingales

1st Edition

By B.G. Ivanoff, Ely Merzbach

Chapman and Hall/CRC

224 pages

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Hardback: 9781584880820
pub: 1999-10-27
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Description

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.

Reviews

"…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

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.

About the Series

Chapman & Hall/CRC Monographs on Statistics and Applied Probability

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Subject Categories

BISAC Subject Codes/Headings:
MAT029000
MATHEMATICS / Probability & Statistics / General
MAT029010
MATHEMATICS / Probability & Statistics / Bayesian Analysis