"A strong feature of the first edition that is further strengthened in the second one is the 'Notes' concluding each chapter. These remarks and discussions not only provide a comprehensive literature review on the historical origins of the material covered in each section, but also point to a variety of recent research papers, where those concepts are applied." (Christian Hirsch, Mathematical Reviews, August, 2018)
1 Random Closed Sets and Capacity Functionals.- 2 Expectations of Random Sets.- 3 Minkowski Sums.- 4 Unions of Random Sets.- 5 Random Sets and Random Functions.- A Topological spaces and metric spaces.- B Linear spaces.- C Space of closed sets.- D Compact sets and the Hausdorff metric.- E Multifunctions and semicontinuity.- F Measures and probabilities.- G Capacities.- H Convex sets.- I Semigroups, cones, and harmonic analysis.- J Regular variation.- References.
Ilya Molchanov is Professor of Probability Theory at the Department of Mathematical Statistics and Actuarial Science at the University of Bern, Switzerland.
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.
