Limit Theorems for Unions of Random Closed Sets

Ilya S. Molchanov

ISBN 10: 3540573933 / ISBN 13: 9783540573937

Published by Springer Berlin Heidelberg, 1993

About this Item

Druck auf Anfrage Neuware - Printed after ordering - The book concerns limit theorems and laws of large numbersfor scaled unionsof independent identically distributedrandom sets. These results generalizewell-known facts fromthe theory of extreme values. Limiting distributions (calledunion-stable) are characterized and found explicitly formany examples of random closed sets. The speed ofconvergence in the limit theorems for unions is estimated bymeans of the probability metrics method.It includes theevaluation of distances between distributions of randomsets constructed similarly to the well-known distancesbetween distributions of random variables. The techniquesinclude regularly varying functions, topological propertiesof the space of closed sets, Choquet capacities, convexanalysis and multivalued functions.Moreover, the concept of regular variation is elaborated formultivalued (set-valued) functions. Applications of thelimit theorems to simulation of random sets, statisticaltests, polygonal approximations of compacts, limit theoremsfor pointwise maxima of random functions are considered.Several open problems are mentioned.Addressed primarily to researchers in the theory of randomsets, stochastic geometry and extreme value theory, the bookwill also be of interest to applied mathematicians workingon applications of extremal processes and their spatialcounterparts. The book is self-contained, and no familiaritywith the theory of random sets is assumed.

Seller Inventory # 9783540573937

Bibliographic Details

• Title: Limit Theorems for Unions of Random Closed Sets
• Publisher: Springer Berlin Heidelberg
• Publication Date: 1993
• Language: English
• ISBN 10: 3540573933
• ISBN 13: 9783540573937
• Binding: Taschenbuch
• Condition: Neu
• Dimensions: 235W x 155H x 10D millimeters 235 millimeters width by 155 millimeters height by 10 millimeters depth
• Weight: 265 grams

About this title

Synopsis

The book concerns limit theorems and laws of large numbers for scaled unionsof independent identically distributed random sets. These results generalizewell-known facts from the theory of extreme values. Limiting distributions (called union-stable) are characterized and found explicitly for many examples of random closed sets. The speed of convergence in the limit theorems for unions is estimated by means of the probability metrics method.It includes the evaluation of distances between distributions of random sets constructed similarly to the well-known distances between distributions of random variables. The techniques include regularly varying functions, topological properties of the space of closed sets, Choquet capacities, convex analysis and multivalued functions. Moreover, the concept of regular variation is elaborated for multivalued (set-valued) functions. Applications of the limit theorems to simulation of random sets, statistical tests, polygonal approximations of compacts, limit theorems for pointwise maxima of random functions are considered. Several open problems are mentioned. Addressed primarily to researchers in the theory of random sets, stochastic geometry and extreme value theory, the book will also be of interest to applied mathematicians working on applications of extremal processes and their spatial counterparts. The book is self-contained, and no familiarity with the theory of random sets is assumed.