A central limit theorem for convex sets

We show that there exists a sequence $\varepsilon_n\searrow0$ for which the following holds: Let K⊂ℝn be a compact, convex set with a non-empty interior. Let X be a random vector that is distributed uniformly in K. Then there exist a unit vector θ in ℝn, t0∈ℝ and σ>0 such that $$\sup_{A\subset\mathbb{R}}\left|\textit{Prob}\,\{\langle X,\theta\rangle\in A\}-\frac{1}{\sqrt{2\pi\sigma}}\int_Ae^{-\frac{(t - t_0)^2}{2\sigma^2}} dt\right|\leq\varepsilon_n,\qquad{(\ast)}$$ where the supremum runs over all measurable sets A⊂ℝ, and where 〈·,·〉 denotes the usual scalar product in ℝn. Furthermore, under the additional assumptions that the expectation of X is zero and that the covariance matrix of X is the identity matrix, we may assert that most unit vectors θ satisfy (*), with t0=0 and σ=1. Corresponding principles also hold for multi-dimensional marginal distributions of convex sets.

[1]  William Feller,et al.  An Introduction to Probability Theory and Its Applications , 1951 .

[2]  I. J. Schoenberg.  On Polya frequency functions , 1951 .

[3]  MATHEMATICSA Property of Logarithmic Concave Functions. II , 1953 .

[4]  C. G. Lekkerkerker.  A Property of Logarithmic Concave Functions. II , 1953 .

[5]  B. Grünbaum.  Partitions of mass-distributions and of convex bodies by hyperplanes. , 1960 .

[6]  Frank E. Grubbs,et al.  An Introduction to Probability Theory and Its Applications , 1951 .

[7]  E. Stein,et al.  Introduction to Fourier Analysis on Euclidean Spaces. , 1971 .

[8]  L. Leindler.  On a Certain Converse of Hölder’s Inequality , 1972 .

[9]  C. Borell.  Convex measures on locally convex spaces , 1974 .

[10]  John R. Rice,et al.  Linear Operators and Approximation , 1974 .

[11]  D. Hensley.  Slicing convex bodies—bounds for slice area in terms of the body’s covariance , 1980 .

[12]  D. Freedman,et al.  Asymptotics of Graphical Projection Pursuit , 1984 .

[13]  V. Milman,et al.  Asymptotic Theory Of Finite Dimensional Normed Spaces , 1986 .

[14]  D. Freedman,et al.  A dozen de Finetti-style results in search of a theory , 1987 .

[15]  Y. Gordon.  Gaussian Processes and Almost Spherical Sections of Convex Bodies , 1988 .

[16]  K. Ball.  Logarithmically concave functions and sections of convex sets in $R^{n}$ , 1988 .

[17]  Y. Gordon.  On Milman's inequality and random subspaces which escape through a mesh in ℝ n , 1988 .

[18]  M. Gromov.  Dimension, non-linear spectra and width , 1988 .

[19]  I. J. Schoenberg.  On Pólya Frequency Functions , 1988 .

[20]  V. Milman,et al.  Isotropic position and inertia ellipsoids and zonoids of the unit ball of a normed n-dimensional space , 1989 .

[21]  G. Schechtman.  A remark concerning the dependence on ɛ in dvoretzky's theorem , 1989 .

[22]  G. Pisier.  The volume of convex bodies and Banach space geometry , 1989 .

[23]  Vitali Milman,et al.  Dvoretzky's theorem — Thirty years later , 1992 .

[24]  D. Stroock,et al.  Probability Theory: An Analytic View , 1995, The Mathematical Gazette.

[25]  Matthieu Fradelizi,et al.  Sections of convex bodies through their centroid , 1997 .

[26]  H. Weizsäcker,et al.  Sudakov's typical marginals, random linear functionals and a conditional central limit theorem , 1997 .

[27]  M. Lifshits,et al.  Average volume of sections of star bodies , 2000 .

[28]  V. Milman,et al.  Chapter 17 - Euclidean Structure in Finite Dimensional Normed Spaces , 2001 .

[29]  J. Lindenstrauss,et al.  Handbook of geometry of Banach spaces , 2001 .

[30]  S. Artstein.  Proportional concentration phenomena on the sphere , 2002 .

[31]  P. Hinow,et al.  Moment inequalities and central limit properties of isotropic convex bodies , 2002 .

[32]  Keith Ball,et al.  The central limit problem for convex bodies , 2003 .

[33]  Sergey G. Bobkov,et al.  On the Central Limit Property of Convex Bodies , 2003 .

[34]  Dan Romik,et al.  Projecting the surface measure of the sphere of ℓpn , 2003 .

[35]  Sergey G. Bobkov,et al.  On concentration of distributions of random weighted sums , 2003 .

[36]  Sanjoy Dasgupta,et al.  An elementary proof of a theorem of Johnson and Lindenstrauss , 2003, Random Struct. Algorithms.

[37]  M. Ledoux.  Spectral gap, logarithmic Sobolev constant, and geometric bounds , 2004 .

[38]  Concentration of mass and central limit properties of isotropic convex bodies , 2004 .

[39]  V. Milman,et al.  Geometry of Log-concave Functions and Measures , 2005 .

[40]  On the (fi sub 2-) behaviour of linear functionals on isotropic convex bodies , 2005 .

[41]  Asymptotic behavior of averages of k-dimensional marginals of measures on Rn , 2005 .

[42]  Concentration of mass on isotropic convex bodies , 2006 .

[43]  G. Paouris.  Concentration of mass on convex bodies , 2006 .

[44]  G. Schechtman.  Two observations regarding embedding subsets of Euclidean spaces in normed spaces , 2006 .

[45]  On Gaussian Marginals of Uniformly Convex Bodies , 2006, math/0604595.

[46]  The Central Limit Problem for Random Vectors with Symmetries , 2005, math/0505618.

[47]  S. Vempala,et al.  The geometry of logconcave functions and sampling algorithms , 2007 .

[48]  The Square Negative Correlation Property for Generalized Orlicz Balls , 2008, 0803.0433.

[49]  S. Sodin.  Tail-Sensitive Gaussian Asymptotics for Marginals of Concentrated Measures in High Dimension , 2005, math/0501382.

[50]  B. Klartag.  Uniform almost sub-Gaussian estimates for linear functionals on convex sets , 2007 .