scholarly journals Asymptotic results for multivariate estimators of the mean density of random closed sets

2016 ◽  
Vol 10 (2) ◽  
pp. 2066-2096 ◽  
Author(s):  
Federico Camerlenghi ◽  
Claudio Macci ◽  
Elena Villa
Keyword(s):  
Asymptotic Results ◽  
Closed Sets ◽  
Random Closed Sets ◽  
The Mean

scholarly journals Variance asymptotics and central limit theorems for volumes of unions of random closed sets

2002 ◽  
Vol 34 (03) ◽  
pp. 520-539 ◽  
Author(s):  
Tomasz Schreiber
Keyword(s):  
Central Limit ◽  
Borel Measure ◽  
Strong Law ◽  
Closed Sets ◽  
Random Closed Sets ◽  
Large Numbers ◽  
Multidimensional Ball ◽  
Mean Width ◽  
The Mean ◽  
Heavy Tailed

Let X, X 1, X 2, … be a sequence of i.i.d. random closed subsets of a certain locally compact, Hausdorff and separable base space E. For a fixed normalised Borel measure μ on E, we investigate the behaviour of random variables μ(E \ (X 1 ∪ ∙ ∙ ∙ ∪ X n )) for large n. The results obtained include a description of variance asymptotics, strong law of large numbers and a central limit theorem. As an example we give an application of the developed methods for asymptotic analysis of the mean width of convex hulls generated by uniform samples from a multidimensional ball. Another example deals with unions of random balls in ℝ d with centres distributed according to a spherically-symmetric heavy-tailed law.

scholarly journals ON THE SPECIFIC AREA OF INHOMOGENEOUS BOOLEAN MODELS. EXISTENCE RESULTS AND APPLICATIONS

2011 ◽  
Vol 29 (2) ◽  
pp. 111 ◽  
Author(s):  
Elena Villa
Keyword(s):  
Measure Theory ◽  
Geometric Measure Theory ◽  
Specific Area ◽  
Boolean Models ◽  
Closed Set ◽  
Geometric Measure ◽  
Closed Sets ◽  
Random Closed Sets ◽  
Random Closed Set ◽  
The Mean

The problem of the evaluation of the so-called specific area of a random closed set, in connection with its mean boundary measure, is mentioned in the classical book by Matheron on random closed sets (Matheron, 1975, p. 50); it is still an open problem, in general. We offer here an overview of some recent results concerning the existence of the specific area of inhomogeneous Boolean models, unifying results from geometric measure theory and from stochastic geometry. A discussion of possible applications to image analysis concerning the estimation of the mean surface density of random closed sets, and, in particular, to material science concerning birth-and-growth processes, is also provided.

scholarly journals Variance asymptotics and central limit theorems for volumes of unions of random closed sets

2002 ◽  
Vol 34 (3) ◽  
pp. 520-539 ◽  
Author(s):  
Tomasz Schreiber
Keyword(s):  
Central Limit ◽  
Borel Measure ◽  
Strong Law ◽  
Closed Sets ◽  
Random Closed Sets ◽  
Large Numbers ◽  
Multidimensional Ball ◽  
Mean Width ◽  
The Mean ◽  
Heavy Tailed

Let X, X1, X2, … be a sequence of i.i.d. random closed subsets of a certain locally compact, Hausdorff and separable base space E. For a fixed normalised Borel measure μ on E, we investigate the behaviour of random variables μ(E \ (X1 ∪ ∙ ∙ ∙ ∪ Xn)) for large n. The results obtained include a description of variance asymptotics, strong law of large numbers and a central limit theorem. As an example we give an application of the developed methods for asymptotic analysis of the mean width of convex hulls generated by uniform samples from a multidimensional ball. Another example deals with unions of random balls in ℝd with centres distributed according to a spherically-symmetric heavy-tailed law.

scholarly journals On the estimation of the mean density of random closed sets

2014 ◽  
Vol 125 ◽  
pp. 65-88 ◽  
Author(s):  
F. Camerlenghi ◽  
V. Capasso ◽  
E. Villa
Keyword(s):  
Closed Sets ◽  
Random Closed Sets ◽  
The Mean

scholarly journals An elementary renewal theorem for random compact convex sets

1995 ◽  
Vol 27 (4) ◽  
pp. 931-942 ◽  
Author(s):  
Ilya S. Molchanov ◽  
Edward Omey ◽  
Eugene Kozarovitzky
Keyword(s):  
Support Function ◽  
Convex Sets ◽  
Compact Convex ◽  
Renewal Function ◽  
Renewal Theorem ◽  
Closed Sets ◽  
Random Closed Sets ◽  
Minkowski Sums ◽  
Image Position ◽  
Unit Vectors

A set-valued analog of the elementary renewal theorem for Minkowski sums of random closed sets is considered. The corresponding renewal function is defined as where are Minkowski (element-wise) sums of i.i.d. random compact convex sets. In this paper we determine the limit of H(tK)/t as t tends to infinity. For K containing the origin as an interior point, where hK(u) is the support function of K and is the set of all unit vectors u with EhA(u) > 0. Other set-valued generalizations of the renewal function are also suggested.

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