Theory of Random Sets

Description

This monograph, now in a thoroughly revised second edition, offers the latest research on random sets. It has been extended to include substantial developments achieved since 2005, some of them motivated by applications of random sets to econometrics and finance. The present volume builds on the foundations laid by Matheron and others, including the vast advances in stochastic geometry, probability theory, set-valued analysis, and statistical inference. It shows the various interdisciplinary relationships of random set theory within other parts of mathematics, and at the same time fixes terminology and notation that often vary in the literature, establishing it as a natural part of modern probability theory and providing a platform for future development. It is completely self-contained, systematic and exhaustive, with the full proofs that are necessary to gain insight. Aimed at research level, Theory of Random Sets will be an invaluable reference for probabilists; mathematicians working in convex and integral geometry, set-valued analysis, capacity and potential theory; mathematical statisticians in spatial statistics and uncertainty quantification; specialists in mathematical economics, econometrics, decision theory, and mathematical finance; and electronic and electrical engineers interested in image analysis. Front Matter ....Pages i-xvi Random Closed Sets and Capacity Functionals (Ilya Molchanov)....Pages 1-223 Expectations of Random Sets (Ilya Molchanov)....Pages 225-316 Minkowski Sums (Ilya Molchanov)....Pages 317-378 Unions of Random Sets (Ilya Molchanov)....Pages 379-449 Random Sets and Random Functions (Ilya Molchanov)....Pages 451-552 Back Matter ....Pages 553-678