## Around Nemirovski’s inequality.(English)Zbl 1355.60010

Banerjee, M. (ed.) et al., From probability to statistics and back: high-dimensional models and processes. A Festschrift in honor of Jon A. Wellner. Including papers from the conference, Seattle, WA, USA, July 28–31, 2010. Beachwood, OH: IMS, Institute of Mathematical Statistics (ISBN 978-0-940600-83-6). Institute of Mathematical Statistics Collections 9, 254-265 (2013).
Let $$X_1,\dots, X_n$$ be (stochastically) independent random variables such that each $$X_i$$ has mean zero, i.e., $$\mathbb{E}(X_i) = 0$$. If $$S_n=\sum_{i=1}^{n}X_i$$, then $\mathbb{E}\;S^2_n =\sum_{i=1}^{n}\mathbb{E}\;X_i^2.$ This can be generalized to vectors in a Hilbert space. If the $$X_i$$’s are independent in a Hilbert space $$\mathcal{H}$$ such that $$\mathbb{E}(X_i) = 0$$ and $$\mathbb{E}\|X_i\|^2 <\infty$$, then $\mathbb{E}\;\|S_n\|^2=\mathbb{E}\;\langle S_n,S_n\rangle=\sum_{i,j}\mathbb{E}\langle X_i,X_j\rangle=\sum_{i}\mathbb{E}\;\|X_i\|^2.$
L. Dümbgen et al. [Am. Math. Mon. 117, No. 2, 138–160 (2010; Zbl 1213.60039)] presented an extension of this inequality for vectors in a Banach space. They gave a constant $$K$$ for which the inequality $\mathbb{E}\;\|S_n\|^2\leq K\;\sum_{i,j}\mathbb{E}\;\|X_i\|^2$ holds true for independent random vectors $$X_i$$ with values in a Banach space.
In the present paper, the authors extend this result by considering $$(\mathbb{E} \|S_n\|)^2$$ instead of $$\mathbb{E}\;\|S_n\|^2$$. Moreover, they consider a finite dimensional Banach space and give a constant $$K$$ in the above inequality which depends on the dimension. The special case of $$\ell^\infty(\mathbb{R}^d)$$ as the Banach space is discussed too.
For the entire collection see [Zbl 1319.62002].

### MSC:

 60B11 Probability theory on linear topological spaces 60E15 Inequalities; stochastic orderings

Zbl 1213.60039
Full Text: