The Central Limit Problem for Random Vectors with Symmetries
Tóm tắt
Motivated by the central limit problem for convex bodies, we study normal approximation of linear functionals of high-dimensional random vectors with various types of symmetries. In particular, we obtain results for distributions which are coordinatewise symmetric, uniform in a regular simplex, or spherically symmetric. Our proofs are based on Stein’s method of exchangeable pairs; as far as we know, this approach has not previously been used in convex geometry. The spherically symmetric case is treated by a variation of Stein’s method which is adapted for continuous symmetries.
Tài liệu tham khảo
Anttila, M., Ball, K., Perissinaki, I.: The central limit problem for convex bodies. Trans. Am. Math. Soc. 355(12), 4723–4735 (2003) (electronic)
Aubert, S., Lam, C.S.: Invariant integration over the unitary group. J. Math. Phys. 44(12), 6112–6131 (2003)
Baldi, P., Rinott, Y., Stein, C.: A normal approximation for the number of local maxima of a random function on a graph. In: Probability, Statistics, and Mathematics: Papers in Honor of Samuel Karlin, pp. 59–81. Academic, Boston (1989)
Ball, K., Perissinaki, I.: The subindependence of coordinate slabs in l n p balls. Israel J. Math. 107, 289–299 (1998)
Bastero, J., Bernués, J.: Asymptotic behaviour of averages of k-dimensional margins of measures on ℝn. Preprint (2005)
Bobkov, S.G.: Personal communication
Bobkov, S.G.: On concentration of distributions of random weighted sums. Ann. Probab. 31(1), 195–215 (2003)
Bobkov, S.G.: Spectral gap and concentration for some spherically symmetric probability measures. In: Geometric Aspects of Functional Analysis (2001–2002). Lecture Notes in Math., vol. 1807, pp. 37–43. Springer, Berlin (2003)
Bobkov, S.G., Koldobsky, A.: On the central limit property of convex bodies. In: Geometric Aspects of Functional Analysis (2001–2002). Lecture Notes in Math., vol. 1807, pp. 44–52. Springer, Berlin (2003)
Brehm, U., Hinow, P., Vogt, H., Voigt, J.: Moment inequalities and central limit properties of isotropic convex bodies. Math. Z. 240(1), 37–51 (2002)
Brehm, U., Voigt, J.: Asymptotics of cross sections for convex bodies. Beitr. Algebra Geom. 41(2), 437–454 (2000)
Bryc, W.: The Normal Distribution, Characterizations with Applications. Lecture Notes in Statistics, vol. 100. Springer, New York (1995)
Diaconis, P., Freedman, D.: A dozen de Finetti-style results in search of a theory. Ann. Inst. H. Poincaré Probab. Stat. 23(2, suppl.), 397–423 (1987)
Gordon, Y.: On Milman’s inequality and random subspaces which escape through a mesh in R n. In: Geometric Aspects of Functional Analysis (1986/87). Lecture Notes in Math., vol. 1317, pp. 84–106. Springer, Berlin (1988)
Holmes, S., Reinert, G.: Stein’s method for the bootstrap. In: Stein’s Method: Expository Lectures and Applications. IMS Lecture Notes Monogr. Ser., vol. 46, pp. 95–136. Inst. Math. Stat., Beachwood (2004)
John, F.: Extremium problems with inequalities as subsidiary conditions. In: Studies and Essays Presented to R. Courant on his 60th Birthday, January 8, 1948, pp. 187–204. Interscience, New York (1948)
Kannan, R., Lovász, L., Simonovits, M.: Isoperimetric problems for convex bodies and a localization lemma. Discrete Comput. Geom. 13(3–4), 541–559 (1995)
Koldobsky, A., Lifshits, M.: Average volume of sections of star bodies. In: Geometric Aspects of Functional Analysis (1996–2000). Lecture Notes in Math., vol. 1745, pp. 119–146. Springer, Berlin (2000)
Ledoux, M.: Spectral gap, logarithmic Sobolev constant, and geometric bounds. In: Surveys in Differential Geometry, vol. ix, pp. 219–240. Int. Press, Somerville (2004)
Ledoux, M.: The Concentration of Measure Phenomenon. Mathematical Surveys and Monographs, vol. 89. American Mathematical Society, Providence (2001)
Meckes, E.: Linear functions on the classical matrix groups. Trans. Am. Math. Soc. (to appear)
Meckes, E.: An infinitesimal version of Stein’s method of exchangeable pairs. Ph.D. thesis, Stanford University (2006)
Naor, A.: The surface measure and cone measure on the sphere of ℓ n p . Trans. Am. Math. Soc. 359(3), 1045–1079 (2007)
Naor, A., Romik, D.: Projecting the surface measure of the sphere of ℓ n p . Ann. Inst. H. Poincaré Probab. Stat. 39(2), 241–261 (2003)
Prékopa, A.: On logarithmic concave measures and functions. Acta Sci. Math. (Szeged) 34, 335–343 (1973)
Reitzner, M.: Central limit theorems for random polytopes. Probab. Theory Relat. Fields 133(4), 483–507 (2005)
Rinott, Y., Rotar, V.: On coupling constructions and rates in the CLT for dependent summands with applications to the antivoter model and weighted U-statistics. Ann. Appl. Probab. 7(4), 1080–1105 (1997)
Schechtman, G., Zinn, J.: On the volume of the intersection of two L n p balls. Proc. Am. Math. Soc. 110(1), 217–224 (1990)
Sodin, S.: Tail-sensitive Gaussian asymptotics for marginals of concentrated measures in high dimension. In: Geometric Aspects of Functional Analysis (2004–2005). Lecture Notes in Math., vol. 1910, pp. 271–295. Springer, Berlin (2007)
Stein, C.: Approximate Computation of Expectations. Institute of Mathematical Statistics Lecture Notes—Monograph Series, vol 7. Institute of Mathematical Statistics, Hayward (1986)
Stein, C.: The accuracy of the normal approximation to the distribution of the traces of powers of random orthogonal matrices. Technical Report 470, Stanford University Dept. of Statistics (1995)
Sudakov, V.N.: Typical distributions of linear functionals in finite-dimensional spaces of high dimension. Dokl. Akad. Nauk SSSR 243(6), 1402–1405 (1978)
Wojtaszczyk, J.O.: The square negative correlation property for generalized Orlicz balls. In: Geometric Aspects of Functional Analysis (2004–2005). Lecture Notes in Math., vol. 1910, pp. 305–313. Springer, Berlin (2007)