An elementary renewal theorem for random compact convex sets

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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Random compact convex sets which are infinitely divisible with respect to Minkowski addition

Advances in Applied Probability ◽

10.2307/1426862 ◽

1979 ◽

Vol 11 (4) ◽

pp. 834-850 ◽

Cited By ~ 11

Author(s):

Shigeru Mase

Keyword(s):

Convex Sets ◽

Compact Convex ◽

Infinite Divisibility ◽

Spectral Measures ◽

Infinitely Divisible ◽

Minkowski Addition ◽

Closed Sets ◽

Random Closed Sets ◽

Canonical Representations ◽

Laplace Transformations

Random closed sets (in Matheron's sense) which are a.s. compact convex and contain the origin are considered. The totality of such random closed sets are closed under the Minkowski addition and we can define the concept of infinite divisibility with respect to Minkowski addition of random compact convex sets. Using a generalized notion of Laplace transformations we get Lévy-type canonical representations of infinitely divisible random compact convex sets. Isotropic and stable cases are also considered. Finally we get several mean formulas of Minkowski functionals of infinitely divisible random compact convex sets in terms of their Lévy spectral measures.

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Random compact convex sets which are infinitely divisible with respect to Minkowski addition

Advances in Applied Probability ◽

10.1017/s0001867800033061 ◽

1979 ◽

Vol 11 (04) ◽

pp. 834-850 ◽

Cited By ~ 10

Author(s):

Shigeru Mase

Keyword(s):

Convex Sets ◽

Compact Convex ◽

Infinite Divisibility ◽

Spectral Measures ◽

Infinitely Divisible ◽

Minkowski Addition ◽

Closed Sets ◽

Random Closed Sets ◽

Canonical Representations ◽

Laplace Transformations

Random closed sets (in Matheron's sense) which are a.s. compact convex and contain the origin are considered. The totality of such random closed sets are closed under the Minkowski addition and we can define the concept of infinite divisibility with respect to Minkowski addition of random compact convex sets. Using a generalized notion of Laplace transformations we get Lévy-type canonical representations of infinitely divisible random compact convex sets. Isotropic and stable cases are also considered. Finally we get several mean formulas of Minkowski functionals of infinitely divisible random compact convex sets in terms of their Lévy spectral measures.

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A large deviation principle for Minkowski sums of heavy-tailed random compact convex sets with finite expectation

Journal of Applied Probability ◽

10.1239/jap/1318940461 ◽

2011 ◽

Vol 48 (A) ◽

pp. 133-144 ◽

Cited By ~ 4

Author(s):

Thomas Mikosch ◽

Zbyněk Pawlas ◽

Gennady Samorodnitsky

Keyword(s):

Convex Sets ◽

Large Deviation ◽

Compact Convex ◽

Convex Compact ◽

Compact Sets ◽

Nonconvex Sets ◽

Minkowski Sums ◽

Heavy Tailed ◽

Regularly Varying Distribution ◽

Convex Compact Sets

We prove large deviation results for Minkowski sums Sn of independent and identically distributed random compact sets where we assume that the summands have a regularly varying distribution and finite expectation. The main focus is on random convex compact sets. The results confirm the heavy-tailed large deviation heuristics: ‘large’ values of the sum are essentially due to the ‘largest’ summand. These results extend those in Mikosch, Pawlas and Samorodnitsky (2011) for generally nonconvex sets, where we assumed that the normalization of Sn grows faster than n.

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Metrizability of Finite Dimensional Spaces with a Binary Convexity

Canadian Journal of Mathematics ◽

10.4153/cjm-1986-001-3 ◽

1986 ◽

Vol 38 (1) ◽

pp. 1-18 ◽

Cited By ~ 1

Author(s):

M. Van de Vel

Keyword(s):

Convex Hull ◽

Weak Topology ◽

Convex Sets ◽

Nonempty Intersection ◽

Finite Collection ◽

Closed Sets ◽

Finite Dimensional ◽

The One ◽

Image Position ◽

Disjoint Convex

A convex structure consists of a set X, together with a collection of subsets of X, which is closed under intersection and under updirected union. The members of are called convex sets, and is a convexity on X. Fox A a subset of X, h (A) denotes the (convex) hull of A. If A is finite, then h(A) is called a polytope, is called a binary convexity if each finite collection of pairwise intersecting convex sets has a nonempty intersection. See [8], [21] for general references.If X is also equipped with a topology, then the corresponding weak topology is the one generated by the convex closed sets. It is usually assumed that at least all polytopes are closed. is called normal provided that for each two disjoint convex closed sets C, D there exist convex closed sets C′, D′, with

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Average distances in compact connected spaces

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972700005827 ◽

1982 ◽

Vol 26 (3) ◽

pp. 331-342 ◽

Cited By ~ 10

Author(s):

David Yost

Keyword(s):

Convex Sets ◽

Average Distance ◽

Compact Convex ◽

Topological Spaces ◽

Normed Spaces ◽

Finite Dimensional ◽

Distance Property ◽

Compact Convex Set ◽

Average Distance Property ◽

Image Position

We give a simple proof of the fact that compact, connected topological spaces have the “average distance property”. For a metric space (X, d), this asserts the existence of a unique number a = a(X) such that, given finitely many points x1, …, xn ∈ X, then there is some y ∈ X withWe examine the possible values of a(X) , for subsets of finite dimensional normed spaces. For example, if diam(X) denotes the diameter of some compact, convex set in a euclidean space, then a(X) ≤ diam(X)/√2 . On the other hand, a(X)/diam(X) can be arbitrarily close to 1 , for non-convex sets in euclidean spaces of sufficiently large dimension.

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Geometric and Topological Properties of Certain w* Compact Convex sets which Arise from the Study of Invariant Means

Canadian Journal of Mathematics ◽

10.4153/cjm-1985-009-9 ◽

1985 ◽

Vol 37 (1) ◽

pp. 107-121 ◽

Cited By ~ 2

Author(s):

Edmond E. Granirer

Keyword(s):

Banach Space ◽

Convex Hull ◽

Fixed Points ◽

Convex Sets ◽

Compact Convex ◽

Topological Properties ◽

Bounded Linear ◽

Invariant Means ◽

Image Position ◽

Set Of Points

Let E be a Banach space, A a subset of its dual E*.x0 ∊ A is said to be a w*Gδ point of A if there are xn ∊ E and scalars γn, n = 1,2, 3 … such thatDenote by w*Gδ{A} the set of all w*Gδ points of A. If S is a semigroup of maps on E* and K ⊂ E*, denote byi.e., the set of points x* in the w*closure of K which are fixed points of S (i.e., sx* = x* for each s in S}. An operator will mean a bounded linear map on a Banach space and Co B will denote the convex hull of B ⊂ E.

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A large deviation principle for Minkowski sums of heavy-tailed random compact convex sets with finite expectation

Journal of Applied Probability ◽

10.1017/s0021900200099186 ◽

2011 ◽

Vol 48 (A) ◽

pp. 133-144 ◽

Cited By ~ 1

Author(s):

Thomas Mikosch ◽

Zbyněk Pawlas ◽

Gennady Samorodnitsky

Keyword(s):

Convex Sets ◽

Large Deviation ◽

Compact Convex ◽

Convex Compact ◽

Compact Sets ◽

Nonconvex Sets ◽

Minkowski Sums ◽

Heavy Tailed ◽

Regularly Varying Distribution ◽

Convex Compact Sets

We prove large deviation results for Minkowski sums S n of independent and identically distributed random compact sets where we assume that the summands have a regularly varying distribution and finite expectation. The main focus is on random convex compact sets. The results confirm the heavy-tailed large deviation heuristics: ‘large’ values of the sum are essentially due to the ‘largest’ summand. These results extend those in Mikosch, Pawlas and Samorodnitsky (2011) for generally nonconvex sets, where we assumed that the normalization of S n grows faster than n.

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Mean Euler characteristic of stationary random closed sets

Stochastic Processes and their Applications ◽

10.1016/j.spa.2021.03.012 ◽

2021 ◽

Vol 137 ◽

pp. 252-271

Author(s):

Jan Rataj

Keyword(s):

Euler Characteristic ◽

Closed Sets ◽

Random Closed Sets

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Crystallochemical classifications of phyllosilicates based on the unified system of projection of chemical composition: I. The mica group

Clay Minerals ◽

10.1180/claymin.1990.025.1.08 ◽

1990 ◽

Vol 25 (1) ◽

pp. 73-81 ◽

Cited By ~ 4

Author(s):

A. Wiewióra

Keyword(s):

Chemical Composition ◽

Divalent Cations ◽

Layer Charge ◽

Vector Representation ◽

Similar Formula ◽

Crystallochemical Formula ◽

Trivalent Cations ◽

Image Position ◽

Composition I ◽

Unit Vectors

AbstractA unified system of vector representation of chemical composition is proposed for the phyllosilicates based on projection of the composition, as given by crystallochemical formula, onto a field with orthogonal axes chosen for octahedral divalent cations, R2+, and Si (X, Y, respectively), and oblique axes for octahedral trivalent cations, R3+, and vacancies, □, (V, Z, respectively). Point coordinates for each set of axes were used to define the direction and length of the unit vectors for phyllosilicates belonging to different groups. Parallel to these fundamental directions the composition isolines were drawn in the projection fields. Applied to micas, this system enables control of the chemical composition by the general crystallochemical formula covering all varieties of Li-free dioctahedral and trioctahedral micas:where z (number of vacancies) = (y-x+ m)/2; m (layer charge) =1; u+y+z = 3. There is a similar formula for vacancy-free lithian micas:where w = m — x+y;m=1; u+y+w = 3, and for Li-free brittle micas:where z = (y — x+m)/2; m = 2; u+y+z = 3. Projection fields were used to classify micas.

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