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A Central Limit Theorem for Convex Chains in the Square

2000
*
Discrete & Computational Geometry
*

An interesting feature is that

doi:10.1007/pl00009490
fatcat:ttgg7d6ikbfnpb3suhanthqwyi
*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. ...##
###
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]

2020
*
arXiv
*
pre-print

Let T be

arXiv:2011.04563v1
fatcat:4drroqdwofaj7nfxplgwfvuyim
*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. ...##
###
ON RANDOM CONVEX CHAINS, ORTHOGONAL POLYNOMIALS, PF SEQUENCES AND PROBABILISTIC LIMIT THEOREMS

2021
*
Mathematika
*

Let T be

doi:10.1112/mtk.12081
fatcat:rtrqwv4czng37cohstfatslcpq
*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 ). ...##
###
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

2012
*
IEEE Transactions on Information Theory
*

Several applications to information theory and

doi:10.1109/tit.2011.2174959
fatcat:ptjdjbcyvrfjhm3zcrpbz27ezi
*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. ...##
###
Random points and lattice points in convex bodies

2008
*
Bulletin of the American Mathematical Society
*

Assume K ⊂ R d is

doi:10.1090/s0273-0979-08-01210-x
fatcat:aprrja6ggfcl7f5upkdugqxcaq
*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. ...##
###
Packing, Hitting, and Coloring Squares
[article]

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. ...

##
###
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]

2011
*
Selected Works of Murray Rosenblatt
*

Acknowledgement: My thanks go to Richard Davis, who made many helpful comments and suggestions and provided

doi:10.1007/978-1-4419-8339-8_2
fatcat:wmedyhbdzffuxgnyuavqj4kwjm
*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.) ...##
###
The Erdős–Nagy theorem and its ramifications

2005
*
Computational geometry
*

Given

doi:10.1016/j.comgeo.2004.12.005
fatcat:kexx6mqeonddxeq6fasy7adebq
*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. ...##
###
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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