Asymptotic behaviour of averages of k-dimensional marginals of measures on ℝⁿ
We study the asymptotic behaviour, as n → ∞, of the Lebesgue measure of the set {x ∈ K ; |PE(x)| ≤ t} for a random kdimensional subspace E ⊂ R and K ⊂ R in the class of isotropic convex bodies. For k growing slowly to infinity, we prove it to be close to the suitably normalised Gaussian measure in R of a t-dilate of the Euclidean unit ball. Some of the results hold for a wider class of probabilities on R. 1. Preliminaries and Notation Let E be a k-dimensional subspace of R, 1 ≤ k ≤ n and denote by PE the orthogonal projection onto E. For any Borel probability P on R, its marginal probability on E is defined as PE(A) := P(A + E⊥) = P(x ∈ R | PE(x) ∈ A), A ⊆ E . A Borel probability P is isotropic if ∫ Rn x dP(x) = 0 and its covariance matrix is a multiple of the identity. A convex body K of volume 1 is isotropic if the uniform measure on K is. In this case, such multiple of the identity is denoted by LK . In [Kl2] the author solved the so called Central limit problem for convex bodies (posed in [ABP], [BV] for k = 1 and considered in [BK], [BHVV], [KL], [MM], [Mi], [Wo]). He showed that every isotropic convex body K (and more generally, every isotropic log-concave probability measure) has the property that most of its k-dimensional marginal distributions are approximately Gaussian, with respect to the total variation metric, provided that k << log n log log n . In a more general probabilistic setting, the k-dimensional version of the problem goes back to [W] (see also [DF], [Bo], [Su]). In [NR], the authors studied proximity of k-marginals to the Gaussian measure with respect to the (weaker) T-distance, for a class of isotropic probabilities satisfying some concentration hypothesis. In [M], Gaussian approximation of k-marginals Date: June 13, 2008. 2000 Mathematics Subject Classification. 52A22, 60F05.