Concentration inequalities for multivariate distributions. II. Elliptically contoured distributions
Michael D. Perlman · Lecture notes-monograph series · 1992
In part I of this study it was shown that Σi Pv x (C) > PΣ 2 (C) under various convexity and symmetry assumptions on the set C C R/\ where PΣ denoted the p-variate normal distribution with mean vector 0 and positive definite covariance matrix Σ.In Part II extensions of these results to the family of elliptically contoured distributions are considered.The proof of the concentration inequality of Fefferman, Jodeit, and Perlman (1972) for convex centrally symmetric sets C is examined to determine whether it can be extended to sets C with other convexity and/or symmetry properties.Whereas it does not appear that this proof remains applicable, in the bivariate case (p = 2) an alternate geometric argument not only extends the concentration inequalities for convex G-invariant sets C and for G-decreasing sets C in Part I to elliptically contoured distributions, but also enlarges the class of groups G for which the concentration inequality for G-decreasing sets is valid.Also, sharpened forms of these concentration inequalities are presented for elliptically contoured distributions that are not absolutely continuous with respect to Lebesgue measure. A Concentration Inequality for Convex Centrally Symmetric SetsIn Part I of this study 2 it was shown that (5.0) Σ!<Σ 2 ^P under various convexity and symmetry assumptions on the set C G It p , where P Σ denoted the p-variate normal distribution with mean vector 0 and positive definite covariance matrix Σ.It is evident that such concentration 1 Research supported in part by National Science Foundation Grant No. DMS-89-02211.AMS 1991 subject classifications.Primary 60E15; Secondary 52A40.Key words and phrases.Multivariate concentration inequalities, elliptically contoured distributions, convex set, group invariance, orthogonal group, cyclic group, dihedral group. 2 Eaton and Perlman (1991).Part I comprised Sections 1-4; Part II comprises Sections 5-7.