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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

Citations:

Zbl 1213.60039
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