Survey on stability of random sets and limit theorems for Minkowski addition

1993 ◽  
pp. 15-27
Author(s):  
Ilya S. Molchanov
Keyword(s):  
Limit Theorems ◽  
Random Sets ◽  
Minkowski Addition

Limit theorems for convex hulls of random sets

1993 ◽  
Vol 25 (02) ◽  
pp. 395-414 ◽  
Author(s):  
Ilya S. Molchanov
Keyword(s):  
Limit Theorems ◽  
Random Sets ◽  
Convex Hulls ◽  
Stable Sets ◽  
Limiting Distributions ◽  
Closed Set ◽  
Closed Sets ◽  
Random Closed Sets ◽  
Random Closed Set ◽  
Simple Limit

Let , be i.i.d. random closed sets in . Limit theorems for their normalized convex hulls conv () are proved. The limiting distributions correspond to C-stable random sets. The random closed set A is called C-stable if, for any , the sets anA and conv ( coincide in distribution for certain positive an, compact Kn , and independent copies A 1, …, An of A. The distributions of C-stable sets are characterized via corresponding containment functionals.

scholarly journals Central limit theorems for fuzzy random sets

2005 ◽  
Vol 15 (3) ◽  
pp. 337-342 ◽  
Author(s):  
Joong-Sung Kwon ◽  
Yun-Kyong Kim ◽  
Sang-Yeol Joo ◽  
Gyeong-Suk Choi
Keyword(s):  
Central Limit ◽  
Limit Theorems ◽  
Central Limit Theorems ◽  
Random Sets ◽  
Fuzzy Random Sets ◽  
Fuzzy Random

Limit Theorems for Markov Random Sets

1973 ◽  
Vol 17 (3) ◽  
pp. 426-433 ◽  
Author(s):  
L. I. Piterbarg
Keyword(s):  
Limit Theorems ◽  
Random Sets ◽  
Markov Random

scholarly journals Limit theorems for stationary tessellations with random inner cell structures

2005 ◽  
Vol 37 (1) ◽  
pp. 25-47 ◽  
Author(s):  
Lothar Heinrich ◽  
Hendrik Schmidt ◽  
Volker Schmidt
Keyword(s):  
Limit Theorems ◽  
Vector Measure ◽  
Individual Cell ◽  
Random Sets ◽  
Point Patterns ◽  
Cell Structures ◽  
Large Numbers ◽  
Sampling Window ◽  
The Individual ◽  
Poisson Type

We consider stationary and ergodic tessellations X = Ξnn≥1 in Rd, where X is observed in a bounded and convex sampling window Wp ⊂ Rd. It is assumed that the cells Ξn of X possess random inner structures, examples of which include point patterns, fibre systems, and tessellations. These inner cell structures are generated, both independently of each other and independently of the tessellation X, by generic stationary random sets that are related to a stationary random vector measure J0 acting on Rd. In particular, we study the asymptotic behaviour of a multivariate random functional, which is determined both by X and by the individual cell structures contained in Wp, as Wp ↑ Rd. It turns out that this functional provides an unbiased estimator for the intensity vector associated with J0. Furthermore, under natural restrictions, strong laws of large numbers and a multivariate central limit theorem of the normalized functional are proven. Finally, we discuss in detail some numerical examples and applications, for which the inner structures of the cells of X are induced by iterated Poisson-type tessellations.

scholarly journals Limit theorems for stationary tessellations with random inner cell structures

2005 ◽  
Vol 37 (01) ◽  
pp. 25-47 ◽  
Author(s):  
Lothar Heinrich ◽  
Hendrik Schmidt ◽  
Volker Schmidt
Keyword(s):  
Limit Theorems ◽  
Vector Measure ◽  
Individual Cell ◽  
Random Sets ◽  
Point Patterns ◽  
Cell Structures ◽  
Large Numbers ◽  
Sampling Window ◽  
The Individual ◽  
Poisson Type

We consider stationary and ergodic tessellations X = Ξ n n≥1 in R d , where X is observed in a bounded and convex sampling window W p ⊂ R d . It is assumed that the cells Ξ n of X possess random inner structures, examples of which include point patterns, fibre systems, and tessellations. These inner cell structures are generated, both independently of each other and independently of the tessellation X, by generic stationary random sets that are related to a stationary random vector measure J 0 acting on R d . In particular, we study the asymptotic behaviour of a multivariate random functional, which is determined both by X and by the individual cell structures contained in W p , as W p ↑ R d . It turns out that this functional provides an unbiased estimator for the intensity vector associated with J 0. Furthermore, under natural restrictions, strong laws of large numbers and a multivariate central limit theorem of the normalized functional are proven. Finally, we discuss in detail some numerical examples and applications, for which the inner structures of the cells of X are induced by iterated Poisson-type tessellations.

Limit Theorems for Exchangeable Random Elements and Random Sets

2000 ◽  
pp. 367-378 ◽  
Author(s):  
Robert L. Taylor ◽  
A. N. Vidyashankar ◽  
Yinpu Chen
Keyword(s):  
Limit Theorems ◽  
Random Sets ◽  
Random Elements

Limit theorems for random compact sets in Banach space

1985 ◽  
Vol 97 (1) ◽  
pp. 151-158 ◽  
Author(s):  
Madan L. Puri ◽  
Dan A. Ralescu
Keyword(s):  
Limit Theorems ◽  
Mathematical Formulation ◽  
General Concept ◽  
Point Of View ◽  
Geometric Probability ◽  
Random Sets ◽  
Compact Sets ◽  
Long Time ◽  
And Control ◽  
First Time

The concept of random set, though vaguely known for a long time (possibly since Buffon's needle problem), did not develop until Robbins [25, 26] provided for the first time a solid mathematical formulation of this concept and investigated relationships between random sets and geometric probabilities. Later on (in a different context) this concept gave rise to a more general concept of set-valued function in topology, and applications were also found in several areas such as economics (see, for example, Aumann[3] and Debreu[10]) and control theory (see, for example, Hermes [13]), among others. Recently in two independent formulations, D. G. Kendall [17] and Matheron[21] provided a comprehensive mathematical theory of this concept influenced by the geometric probability point of view. Actually, much of the research in this area falls under the heading of stochastic geometry (see, for example, M. G. Kendall and P. A. P. Moran [18]). The D. G. Kendall and Matheron theories have been compared and ‘reconciled’ by Ripley [24]. In the past few years random sets have been investigated as extensions of random variables and random vectors, and in this framework the problems of deriving limit theorems have received a great deal of attention. This approach is benefiting greatly from probability results in Banach spaces.

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