Asymptotic behaviour of averages of $k$-dimensional marginals of measures on $\mathbb R^n$

Jesús Bastero ¹   ; Julio Bernués ¹

¹ Departamento de Matemáticas Universidad de Zaragoza 50009 Zaragoza, Spain

Studia Mathematica, Tome 190 (2009) no. 1, pp. 1-31

Résumé

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 $\mathbb R^n$.

Détail

DOI : 10.4064/sm190-1-1

Keywords: study asymptotic behaviour infty lebesgue measure set vert vert random k dimensional subspace subset mathbb isotropic convex body subset mathbb growing slowly infinity prove close suitably normalised gaussian measure mathbb t dilate euclidean unit ball results wider class probabilities mathbb