# Dimension reduction for low-order moments of Rademacher-weighted sums of vectors

Let $x_1,\dots,x_n$ be vectors in a Euclidean space $H$. Let $\varepsilon_1,\dots,\varepsilon_n$ be independent Rademacher random variables (r.v.’s), so that $P(\varepsilon_i=\pm1)=1/2$ for all $i$.
Does the inequality
$$(1)\qquad E f\Big(\Big\|\sum_{i=1}^n\varepsilon_i x_i\Big\|\Big) \le E f\Big(\sum_{i=1}^n\varepsilon_i \|x_i\|\Big)$$
always hold if $f(x)=f_p(x):=|x|^p$ for some real $p\ge2$ and all real $x$? For $f=f_p$ with $p\ge3$, this inequality is due to Utev .

As noted in , it is enough to prove $(1)$ for $f=g_a$, where $a\ge0$ and $g_a(x):=\max(0,|x|-a)^2$ for all real $x$. It is also shown in  (where more context can be found) that one may assume without loss of generality that the dimension of the Euclidean space $H$ is no greater than $\left\lfloor(\sqrt{1+8n}-1)/2\right\rfloor\sim\sqrt{2n}$.

One corollary of (1) would be the extension of known exact Rosenthal-type bounds [1,3,4] for independent symmetric real-valued r.v.’s to independent symmetric random vectors. Indeed, let now $H$ be a separable Hilbert space, and let $X_1,\dots,X_n$ be independent symmetric random vectors in $H$. Conditioning on the random sets $\{X_i,-X_i\}$ and using Theorem 5.2 in  or formula (3) in , one sees that the conjectured inequality (1) above with $f=f_p$ would yield
$$E\Big\|\sum_{i=1}^n X_i\Big\|^p\le\sum_{i=1}^n E\|X_i\|^p+E|Z|^p \Big(\sum_{i=1}^n E\|X_i\|^2\Big)^{p/2}$$
for $p\in(2,4]$, and this upper bound on $E\Big\|\sum_{i=1}^n X_i\Big\|^p$ is exact in terms of $\sum_{i=1}^n E\|X_i\|^p$ and $\sum_{i=1}^n E\|X_i\|^2$; here $Z$ is a standard normal r.v.

Quite similarly one can obtain the exact upper bound on $E\Big\|\sum_{i=1}^n X_i\Big\|^p$ for $p>4$ (again in the “symmetric” case), say by using Theorem 5 in  or formula (6) in ; recall that for $f=f_p$ with $p\ge3$ inequality (1) is known.

References

 Utev, S.A., Extremal problems in moment inequalities. (Russian) Limit theorems of probability theory, 56–75, 175,
Trudy Inst. Mat., 5, “Nauka” Sibirsk. Otdel., Novosibirsk, 1985; MathSciNet Review MR0821753.

 Pinelis, I., Dimensionality reduction in extremal problems for moments of linear combinations of vectors with random coefficients. Stochastic inequalities and applications, 169–185, Progr. Probab., 56, Birkhäuser, Basel, 2003; MathSciNet Review MR2073433.

 Figiel, T.; Hitczenko, P.; Johnson, W. B.; Schechtman, G.; Zinn, J.
Extremal properties of Rademacher functions with applications to the Khintchine and Rosenthal inequalities.
Trans. Amer. Math. Soc. 349 (1997), no. 3, 997–1027; MathSciNet Review MR1390980.

 Ibragimov, R.; Sharakhmetov, Sh. On an exact constant for the Rosenthal inequality. Theory Probab. Appl. 42 (1997), no. 2, 294–302; MathSciNet Review MR1474714.