Nemirovski's inequality states that given independent and centered at expectation random vectors $X_{1},\ldots,X_{n}$ with values in $\ell^p(\mathbb{R}^d)$, there exists some constant $C(p,d)$ such that \[\mathbb{E}\Vert S_n\Vert _p^2\le C(p,d)\sum_{i=1}^{n}\mathbb{E}\Vert X_i\Vert _p^2.\] Furthermore $C(p,d)$ can be taken as $\kappa(p\wedge \log(d))$. Two cases were studied further in [ Am. Math. Mon. 117(2) (2010) 138–160]: general finite-dimensional Banach spaces and the special case $\ell^{\infty}(\mathbb{R}^{d})$. We show that in these two cases, it is possible to replace the quantity $\sum_{i=1}^n\mathbb{E}\Vert X_i\Vert _p^2$ by a smaller one without changing the order of magnitude of the constant when $d$ becomes large. In the spirit of [ Am. Math. Mon. 117(2) (2010) 138–160], our approach is probabilistic. The derivation of our version of Nemirovski's inequality indeed relies on concentration inequalities.
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