Asymptotic behaviour of averages of $k$-dimensional marginals of measures on $\mathbb R^n$
Volume 190 / 2009
Studia Mathematica 190 (2009), 1-31
MSC: 53A20, 60F05.
DOI: 10.4064/sm190-1-1
Abstract
We study the asymptotic behaviour, as $n\to\infty$, of the Lebesgue measure of the set $ \{x\in K: \vert P_E(x)\vert\le t\}$ for a random $k$-dimensional subspace $E\subset\mathbb R^n$ and an isotropic convex body $K\subset\mathbb R^n$. For $k$ growing slowly to infinity, we prove it to be close to the suitably normalised Gaussian measure in $\mathbb R^k$ of a $t$-dilate of the Euclidean unit ball. Some of the results hold for a wider class of probabilities on $\mathbb R^n$.
