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A Central Limit Theorem for Convex Chains in the Square

I. Bárány, G. Rote, W. Steiger, C.-H. Zhang
2000 Discrete & Computational Geometry  
An interesting feature is that the limit shape is a direct consequence of the method. The main result is an accompanying central limit theorem for these chains.  ...  . , P n in the unit square define a convex n-chain if they are below y = x and, together with P 0 = (0, 0) and P n+1 = (1, 1), they are in convex position.  ...  This variable was studied in Sinai's paper [7] where, in addition, a central limit theorem for the deviations between P and L was stated.  ... 
doi:10.1007/pl00009490 fatcat:ttgg7d6ikbfnpb3suhanthqwyi

Page 4564 of Mathematical Reviews Vol. , Issue 89H [page]

1989 Mathematical Reviews  
Book (1-CASDH) 89h:60038 60F05 60G42 Haeusler, Erich (D-MNCH); Joos, Konrad (D-MNCH) A nonuniform bound on the rate of convergence in the martingale central limit theorem. Ann.  ...  Resnick (1-CRNL-O) 89h:60037 60F05S 62G30 Dzhamirzaev, A. A. (2-TASH); Mamurov, I. N. A shift theorem for the central terms of an order statistics sequence. (Russian) Dokl. Akad.  ... 

Page 533 of The Annals of Mathematical Statistics Vol. 31, Issue 2 [page]

1960 The Annals of Mathematical Statistics  
A Central Limit Theorem for Systems of Regressions. E. J. Hannan, University of North Carolina. (Introduced by David B.  ...  It is shown that this theorem does not extend directly to higher dimen- sions: namely, that in three dimensions, there are convex sets symmetric with respect to the origin which cannot be obtained as a  ... 

Page 1881 of Mathematical Reviews Vol. , Issue 2001C [page]

2001 Mathematical Reviews  
The main result is a CLT for these chains. A weak convergence result implies several other statements concerning deviation between random convex chains and their limit.  ...  For these general po-monoids the author generalizes all the usual notions of probability and proves the weak law of large numbers and the central limit theorem.  ... 

On random convex chains, orthogonal polynomials, PF sequences and probabilistic limit theorems [article]

Anna Gusakova, Christoph Thäle
2020 arXiv   pre-print
Let T be the triangle in the plane with vertices (0,0), (0,1) and (0,1). The convex hull of (0,1), (1,0) and n independent random points uniformly distributed in T is the random convex chain T_n.  ...  A three-term recursion for the probability generating function G_n of the number f_0(T_n) of vertices of T_n is proved.  ...  Acknowledgement This work has been supported by the Deutsche Forschungsgemeinschaft (DFG) via SPP 2265 Random Geometric Systems.  ... 
arXiv:2011.04563v1 fatcat:4drroqdwofaj7nfxplgwfvuyim

ON RANDOM CONVEX CHAINS, ORTHOGONAL POLYNOMIALS, PF SEQUENCES AND PROBABILISTIC LIMIT THEOREMS

Anna Gusakova, Christoph Thäle
2021 Mathematika  
Let T be the triangle in the plane with vertices (0,0), (0,1) and (0,1). The convex hull of (0,1), (1,0) and n independent random points uniformly distributed in T is the random convex chain T n .  ...  Therefore, if T denotes the triangle with corners at (0,0), (0,1) and (1,0), one considers the convex chain T n , which is the convex hull of n ∈ N uniform random points in T together with the corners  ...  in a polygon P together with a marked corner illustrating the reduction to a random convex chain (left) and a random convex chain T n in the right triangle T (right). distribution of f 0 (T n ).  ... 
doi:10.1112/mtk.12081 fatcat:rtrqwv4czng37cohstfatslcpq

Page 2034 of Mathematical Reviews Vol. 53, Issue 6 [page]

1977 Mathematical Reviews  
A. 14603 On the accuracy of the approximation in the central limit theorem. (Russian. English summary) Teor. Verojatnost. i Primenen. 21 (1976), no. 1, 107-122.  ...  V. 14595 A description of multidimensional limit laws for finite Markov chains in a scheme of arrays. The general case. (Russian. English summary) Teor. Verojatnost. i Mat. Statist.  ... 

Functional Properties of Minimum Mean-Square Error and Mutual Information

Yihong Wu, Sergio Verdu
2012 IEEE Transactions on Information Theory  
Several applications to information theory and the central limit theorem are discussed.  ...  Index Terms-Bayesian statistics, central limit theorem, Gaussian noise, minimum mean-square error (MMSE), mutual information, non-Gaussian noise.  ...  ACKNOWLEDGMENT The authors are grateful to Associate Editor Dongning Guo for a careful reading of the draft and many fruitful discussions.  ... 
doi:10.1109/tit.2011.2174959 fatcat:ptjdjbcyvrfjhm3zcrpbz27ezi

Random points and lattice points in convex bodies

Imre Bárány
2008 Bulletin of the American Mathematical Society  
Assume K ⊂ R d is a convex body and X is a (large) finite subset of K. How many convex polytopes are there whose vertices belong to X? Is there a typical shape of such polytopes?  ...  How well does the maximal such polytope (which is actually the convex hull of X) approximate K? We are interested in these questions mainly in two cases.  ...  Acknowledgments Part of this paper was written on a pleasant and fruitful visit by the author to the Institute for Advanced Study at the Hebrew University in Jerusalem.  ... 
doi:10.1090/s0273-0979-08-01210-x fatcat:aprrja6ggfcl7f5upkdugqxcaq

Packing, Hitting, and Coloring Squares [article]

Marco Caoduro, András Sebő
2022 arXiv   pre-print
For comparison, in the axis-parallel case, the supremum of the considered ratio is in the interval [3/2,2] for unit squares and [3/2,4] for arbitrary squares.  ...  Given a family of squares in the plane, their packing problem asks for the maximum number, ν, of pairwise disjoint squares among them, while their hitting problem asks for the minimum number, τ, of points  ...  Conversely, it is also true that any centrally symmetric compact convex set is the unit ball for a norm.  ... 
arXiv:2206.02185v2 fatcat:zkghud5i6zeddmlzp6ccsmwsbq

Page 2722 of Mathematical Reviews Vol. , Issue 93e [page]

1993 Mathematical Reviews  
central limit theorem to be true.  ...  Summary: “We discuss a uniform central limit theorem for set- indexed partial sum processes when the second moment is not assumed to be finite, and give a necessary and sufficient condition for the uniform  ... 

Murray Rosenblatt's Contributions to Strong Mixing [chapter]

Richard A. Davis, Keh-Shin Lii, Dimitris N. Politis
2011 Selected Works of Murray Rosenblatt  
Acknowledgement: My thanks go to Richard Davis, who made many helpful comments and suggestions and provided the third paragraph of this article.  ...  Central limit theory under the p-mixing condition apparently started with central limit theorems of Rosenblatt [19] [20, Chapter 7] for square-integrable "instantaneous" functions of strictly stationary  ...  central limit theorem for random sequences.)  ... 
doi:10.1007/978-1-4419-8339-8_2 fatcat:wmedyhbdzffuxgnyuavqj4kwjm

The Erdős–Nagy theorem and its ramifications

Godfried Toussaint
2005 Computational geometry  
Given a simple polygon in the plane, a flip is defined as follows: consider the convex hull of the polygon. If there are no pockets do not perform a flip.  ...  Several results in the literature are improved with the application of the theorem.  ...  Acknowledgements I would like to thank Jorge Calvo, Erik Demaine, Joe O'Rourke and Branko Grünbaum for reading the manuscript and providing constructive comments.  ... 
doi:10.1016/j.comgeo.2004.12.005 fatcat:kexx6mqeonddxeq6fasy7adebq

Page 543 of Mathematical Reviews Vol. , Issue 2000a [page]

2000 Mathematical Reviews  
(PL-TORN; Torun) A central limit theorem for strictly stationary sequences in terms of slow variation in the limit. (English summary) Probab. Math. Statist. 18 (1998), no. 2, 359-368.  ...  The author constructs a co-valued random variable such that (S,/./n) has a convergent subsequence, but X does not satisfy the central limit theorem.  ... 

Page 2196 of Mathematical Reviews Vol. 52, Issue 6 [page]

1976 Mathematical Reviews  
In Section 7 central limit theorem for the stochastic process {X(t)} is obtained and in Section 8 the Fourier series expansion for harmonizable processes is considered.  ...  Les auteurs proposent une nouvelle démonstration du théoréme limite centrale pour des chaines a liaisons complétes 4 un nombre fini d’états. {For the entire collection see MR 52 #9308.}  ... 
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