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Choquet–Kendall–Matheron theorems for non-Hausdorff spaces

JEAN GOUBAULT-LARRECQ, KLAUS KEIMEL
2011 Mathematical Structures in Computer Science  
We establish Choquet-Kendall-Matheron theorems on non-Hausdorff topological spaces.  ...  We obtain three variants of the theorem, each one characterizing distributions, in the form of continuous valuations, over relevant powerdomains of demonic, resp. angelic, resp. erratic non-determinism  ...  We thank Achim Jung for giving us the opportunity to do so. We thank the anonymous referee for suggesting the actual simplified version of the proof of Proposition 4.11.  ... 
doi:10.1017/s0960129510000617 fatcat:ppzbijmjvbfdbp5exndhsy7pey

On metrization of the hit-or-miss topology using Alexandroff compactification

Guo Wei, Yangeng Wang
2007 International Journal of Approximate Reasoning  
When the hit-or-miss topology is employed, the space of all closed subsets of a Hausdorff, locally compact and second countable space (HLCSC) is known to be Hausdorff, compact and second countable, thus  ...  With the constructed metric, we investigate a hyperspace Birkhoff ergodic theorem to explore the connection between orbital behaviors of hyperspace dynamical systems and Choquet capacities of random closed  ...  We thank the referee for pointing out to us this previous significant work on the case R n . We also thank Professor Hung T. Nguyen for calling our attention to the Hausdorff-Buseman metric.  ... 
doi:10.1016/j.ijar.2006.12.007 fatcat:s7f6pdos55gdhjfc22ovrpu64q

On Means and Their Asymptotics: Circles and Shape Spaces

Stephan Huckemann, Thomas Hotz
2013 Journal of Mathematical Imaging and Vision  
We review the statistical role of the cut locus of intrinsic means for circles as well as that of singular strata for shape spaces (occurring where the group action is degenerate) and conclude with an  ...  E.g. in geodesic principal component analysis for shape spaces, shape data are described by generalized geodesics which naturally form a shape space of their own, different from the original one.  ...  For our purpose here, we rely on the definition of random closed sets as introduced and studied by Choquet (1954) , Kendall (1974) and Matheron (1975) .  ... 
doi:10.1007/s10851-013-0462-3 fatcat:vmnbnzd46ve4dlrxmxkfp2dckm

Page 968 of Mathematical Reviews Vol. 52, Issue 3 [page]

1976 Mathematical Reviews  
Topologies PROBABILITY THEORY AND STOCHASTIC PROCESSES are constructed on the spaces of closed, open or compact subsets of a locally compact separable Hausdorff space which enable the assignment of probabilities  ...  D. 6824 Limit theorems for compositions of distributions on certain nil- potent Lie groups. (Russian. English summary) Teor. Verojatnost. i Primenen. 19 (1974), 84-103.  ... 

Characterizing joint distributions of random sets by multivariate capacities

Bernhard Schmelzer
2012 International Journal of Approximate Reasoning  
For the special case of finite spaces a multivariate version of the Moebius inversion formula is derived.  ...  By the Choquet theorem, distributions of random closed sets can be characterized by a certain class of set functions called capacity functionals.  ...  Acknowledgements I would like to thank the reviewers (of this paper and paper [21] ) for their useful comments and suggestions.  ... 
doi:10.1016/j.ijar.2012.06.017 fatcat:lr72oty2u5fitmsh5blxtvtkti

Intrinsic Inference on the Mean Geodesic of Planar Shapes and Tree Discrimination by Leaf Growth [article]

Stephan Huckemann
2010 arXiv   pre-print
For planar landmark based shapes, taking into account the non-Euclidean geometry of the shape space, a statistical test for a common mean first geodesic principal component (GPC) is devised.  ...  For both scenarios, strong consistency and central limit theorems are established, along with an algorithm for the computation of a Ziezold mean geodesic.  ...  For our purpose here, we rely on the definition of random closed sets as introduced and studied by Choquet (1954) , Kendall (1974) and Matheron (1975) .  ... 
arXiv:1009.3203v1 fatcat:gxih2g4uhbacratql4jmdu5thm

On the meaning of mean shape: manifold stability, locus and the two sample test

Stephan F. Huckemann
2012 Annals of the Institute of Statistical Mathematics  
In particular, for non-manifolds, such as Kendall's 3D shape space, this paper answers the question, for which means one may apply a two-sample test.  ...  The underlying general result of manifold stability of a mean on a shape space, the quotient due to an proper and isometric action of a Lie group on a Riemannian manifold, blends the slice theorem from  ...  Acknowledgments The author would like to thank Alexander Lytchak for helpful advice on differential geometric issues. Also, the author gratefully acknowledges support by DFG Grant HU 1575/2-1.  ... 
doi:10.1007/s10463-012-0352-2 fatcat:yvy2y5uajremjot3jr7uxjpy6i

Intrinsic inference on the mean geodesic of planar shapes and tree discrimination by leaf growth

Stephan F. Huckemann
2011 Annals of Statistics  
For planar landmark based shapes, taking into account the non-Euclidean geometry of the shape space, a statistical test for a common mean first geodesic principal component (GPC) is devised which rests  ...  For both scenarios, strong consistency and central limit theorems are established, along with an algorithm for the computation of a Ziezold mean geodesic.  ...  Moreover, he is indebted to Thomas Hotz for discussing statistical issues and to David Glickenstein for a comment on geometric aspects.  ... 
doi:10.1214/10-aos862 fatcat:6gj7vugrbbehdc2yksynfmn3ou

Integration in a dynamical stochastic geometric framework

Giacomo Aletti, Enea G. Bongiorno, Vincenzo Capasso
2011 E S A I M: Probability & Statistics  
In this framework, growth is generally anisotropic and, according to a mesoscale point of view, it is not local, i.e. for a fixed time instant, growth is the same at each space point.  ...  We propose a set-valued framework for the well-posedness of birth-and-growth process.  ...  The well known Choquet-Kendall-Matheron Theorem states that, the probability law P X of any RaCS X is uniquely determined by its hitting function (see [26] ) and hence by Q X (K) = 1 − T X (K).  ... 
doi:10.1051/ps/2010009 fatcat:aqksudnuivgvpgnrvsag5b2izy

On the meaning of mean shape [article]

Stephan Huckemann
2011 arXiv   pre-print
In particular for non-manifolds such as Kendall's 3D shape space, this paper answers the question, for which means one may apply a two-sample test.  ...  The underlying general result of manifold stability of a mean on a shape space, the quotient due to an isometric action of a compact Lie group on a Riemannian manifold, blends the Slice Theorem from differential  ...  Acknowledgment The author would like to thank Alexander Lytchak for helpful advice on differential geometic issues.  ... 
arXiv:1002.0795v2 fatcat:zjepgjwysraszeomeek2e27sfe

Quermass-interaction processes: conditions for stability

W. S. Kendall, M. N. M. van Lieshout, A. J. Baddeley
1999 Advances in Applied Probability  
The main question addressed by the paper is that of when the resulting point process is well-defined: geometric arguments are used to establish conditions for the point process to be stable in the sense  ...  We are also grateful for the helpful remarks of an anonymous referee.  ...  Acknowledgements This work arose from a visit to CWI Amsterdam by WSK, who gratefully acknowledges the support of CWI for this visit.  ... 
doi:10.1239/aap/1029955137 fatcat:g7xywmz5g5eqvdb25vfp5i335a

Quermass-interaction processes: conditions for stability

W. S. Kendall, M. N. M. van Lieshout, A. J. Baddeley
1999 Advances in Applied Probability  
The main question addressed by the paper is that of when the resulting point process is well-defined: geometric arguments are used to establish conditions for the point process to be stable in the sense  ...  We are also grateful for the helpful remarks of an anonymous referee.  ...  Acknowledgements This work arose from a visit to CWI Amsterdam by WSK, who gratefully acknowledges the support of CWI for this visit.  ... 
doi:10.1017/s0001867800009137 fatcat:rlziir7mjngspdt2bgj6xcmsti

The spans in Brownian motion

Steven Evans, Jim Pitman, Wenpin Tang
2017 Annales de l'I.H.P. Probabilites et statistiques  
For d ∈{1,2,3}, let (B^d_t; t ≥ 0) be a d-dimensional standard Brownian motion. We study the d-Brownian span set Span(d):={t-s; B^d_s=B^d_t for some 0 ≤ s ≤ t}.  ...  In addition, we show that Span(1)=R_+ almost surely; the Lebesgue measure of Span(2) is 0 almost surely and its Hausdorff dimension is 1 almost surely; and the Hausdorff dimension of Span(3) is 1/2 almost  ...  Acknowledgements We thank Alexander Holroyd for pointing out the direct proof of Theorem 1.1 (1) . We also thank Yuval Peres for helpful discussions.  ... 
doi:10.1214/16-aihp749 fatcat:q4bumpqcjjcztjpbx4z7n32u6m

Random Closed Sets [chapter]

I. Molchanov
Space, Structure and Randomness  
This survey highlights major contributions of George Matheron to developments of random sets theory.  ...  It outlines the state of the art of this theory before Matheron, other approaches parallel to Matheron's work and mentions some of more recent developments.  ...  Theorem 2 (Choquet-Kendall-Matheron theorem). Let T : K → [0, 1].  ... 
doi:10.1007/0-387-29115-6_7 fatcat:oizqxbsazjf5xnhmkdapujttiu

Statistical aspects of fuzzy monotone set-valued stochastic processes. Application to birth-and-growth processes

Giacomo Aletti, Enea G. Bongiorno, Vincenzo Capasso
2009 Fuzzy sets and systems (Print)  
A decomposition theorem is established to characterize the nucleation and the growth. As a consequence, different consistent set--valued estimators are studied for growth process.  ...  The well known Choquet-Kendall-Matheron Theorem states that, the probability law P X of any RaCS X is uniquely determined by its hitting function (see [21] ) and hence by Q X (K) = 1 − T X (K).  ...  For any A, B ∈ F ′ the Hausdorff distance (or metric) is defined by δ H (A, B) = max sup a∈A inf b∈B a − b X , sup b∈B inf a∈A a − b X .  ... 
doi:10.1016/j.fss.2008.12.011 fatcat:3j4octzuwbbqbct5bfr6vl7lfm
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