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 [1].

As noted in [2], 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 [2] (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 [3] or formula (3) in [4], 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 [1] or formula (6) in [4]; recall that for $f=f_p$ with $p\ge3$ inequality (1) is known.

References

[1] 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.

[2] 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.

[3] 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.

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

Answer

Attribution
Source : Link , Question Author : Iosif Pinelis , Answer Author : Community

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