### A characterization of the domain of attraction of a normal distribution in a Hilbert space

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The present article studies the conditions under which the almost everywhere convergence and the convergence in measure coincide. An application in the statistical estimation theory is outlined as well.

The Riesz transforms of a positive singular measure $\nu \in M\left({\mathbf{R}}^{n}\right)$ satisfy the weak type inequality$$m\left[\sum _{j=1}^{n}|{R}_{j}\nu |\>\lambda \right]\ge \frac{C\parallel \nu \parallel}{\lambda},\phantom{\rule{3.33333pt}{0ex}}\lambda \>0$$where $m$ denotes Lebesgue measure and $C$ is a positive constant only depending on $m$.

We prove that there exist constants C>0 and 0 < λ < 1 so that for all convex bodies K in ${\mathbb{R}}^{n}$ with non-empty interior and all integers k so that 1 ≤ k ≤ λn/ln(n+1), there exists a k-dimensional affine subspace Y of ${\mathbb{R}}^{n}$ satisfying $d(Y\cap K,{B}_{2}^{k})\le C(1+\surd (k/ln(n/\left(kln(n+1)\right)))$. This formulation of Dvoretzky’s theorem for large dimensional sections is a generalization with a new proof of the result due to Milman and Schechtman for centrally symmetric convex bodies. A sharper estimate holds for the n-dimensional simplex.