Abstract
In this note, we study possible extensions of the Central Limit Theorem for non-convex bodies. First, we prove a Berry-Esseen type theorem for a certain class of unconditional bodies that are not necessarily convex. Then, we consider a widely-known class of non-convex bodies, the so-called p-convex bodies, and construct a counter-example for this class.
Keywords
- Classical Central Limit Theorem
- Unconditional Bodies
- Berry-Esseen Type Theorem
- Constant Density Function
- Haar Probability Measure
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This is a preview of subscription content, access via your institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsReferences
M. Anttila, K. Ball, I. Perissinaki, The central limit problem for convex bodies. Trans. Am. Math. Soc. 355, 4723–4735 (2003)
J. Bastero, J.Bernues, A.Pena, An extension of Milman’s reverse Brunn-Minkowski inequality. Geom. Funct. Anal. 5(3), 572–581 (1995)
G. Bennet, Probability inequalities for the sum of independent random variables. J. Am. Stat. Assoc. 57, 33–45 (1962)
L. Berwald, Verallgemeinerung eines Mittelwertsatzes von J. Favard für positive konkave Funktionen. Acta Math. 79, 17–37 (1947)
C. Borell, Convex measures on locally convex spaces. Arkiv för Matematik 12(1–2), 239–252 (1974)
U. Brehm, J. Voigt, Asymptotics of cross sections for convex bodies. Beiträge Algebra Geom. 41, 437–454 (2000)
S.J. Dilworth, The dimension of Euclidean subspaces of quasinormed spaces. Math. Proc. Camb. Philos. Soc. 97(2), 311–320 (1985)
W. Feller, An Introduction to Probability Theory and its Applications, vol. II, Sect. XVI.5 (Wiley, New York, 1971)
Y. Gordon, N.J. Kalton, Local structure theory for quasi-normed spaces. Bull. Sci. Math. 118, 441–453 (1994)
Y. Gordon, D.R. Lewis, Dvoretzky’s theorem for quasi-normed space. Illinois J. Math. 35(2), 250–259 (1991)
N.J. Kalton, Convexity, type and the three space problem. Stud. Math. 69, 247–287 (1981)
B. Klartag, A central limit theorem for convex sets. Invent. Math. 168, 91–131 (2007)
B. Klartag, A Berry-Esseen type inequality for convex bodies with an unconditional basis. Probab. Theory Relat. Fields 45(1), 1–33 (2009)
B. Klartag, Poincaré inequalities and moment maps. Ann. Fac. Sci. Toulouse Math. 22(1), 1–41 (2013)
A.E. Litvak, Kahane-Khinchin’s inequality for the quasi-norms. Can. Math. Bull. 43, 368–379 (2000)
A.E. Litvak, V.D. Milman, N. Tomczak-Jaegermann, Isomorphic random subspaces and quotients of convex and quasi-convex bodies, in GAFA. Lecture Notes in Mathematics, vol. 1850 (Springer, Berlin, 2004), pp. 159–178
V. Milman, Isomorphic Euclidean regularization of quasi-norms in Rn. C. R. Acad. Sci. Paris 321(7), 879–884 (1995)
V.D. Milman, A. Pajor, Isotropic position and inertia ellipsoids and zonoids of the unit ball of a normed n-dimensional space, in Geometric Aspects of Functional Analysis - Israel Seminar. Lecture Notes in Mathematics, vol. 1376 (Springer, Berlin, 1989), pp. 64–104
Acknowledgements
This paper is part of the authors M.Sc. thesis written under the supervision of Professor Bo’az Klartag whose guidance, support and patience were invaluable. In addition, I would like to thank Andrei Iacob for his helpful editorial comments. Supported by the European Research Council (ERC).
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Grupel, U. (2014). Remarks on the Central Limit Theorem for Non-convex Bodies. In: Klartag, B., Milman, E. (eds) Geometric Aspects of Functional Analysis. Lecture Notes in Mathematics, vol 2116. Springer, Cham. https://doi.org/10.1007/978-3-319-09477-9_14
Download citation
DOI: https://doi.org/10.1007/978-3-319-09477-9_14
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-09476-2
Online ISBN: 978-3-319-09477-9
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)
