19
$\begingroup$

Let $a_1,\dots,a_n$ be real numbers such that $a_1^2+\dots+a_n^2=1$. Let $\eta_1,\dots,\eta_n$ be independent Rademacher random variables (r.v.'s), so that $P(\eta_i=\pm1)=\frac12$ for all $i$. Let $S:=a_1\eta_1+\dots+a_n\eta_n$, and let $T$ be an independent copy of $S$. Does then the inequality $$(*)\qquad E f_p(S)\le E f_p\Big(\frac{S+T}{\sqrt2}\Big)$$ hold for all real $p\ge2$, where $f_p(x):=|x|^p$?

This conjecture was communicated to me by Joel Zinn quite some time ago, and I think it deserves to be more broadly known. A motivation behind it was to obtain an alternative (and hopefully easier) proof of Haagerup's inequality $$E f_p(S)\le E f_p(Z)$$ Haagerup, which indeed easily follows from $(*)$ by the central limit theorem; here $Z$ is a standard normal r.v. For $p=2$, $(*)$ is trivial. For $p\ge3$, $(*)$ is easily proved; in particular, it follows immediately from Corollary 2.5 in T^2.

So, actually the question is only about $p\in(2,3)$. In that case, in view of Lemma 2.2 in Figiel et al, it would be enough to prove $(*)$ with $g_t$ in place of $f_p$ (for all real $t>0$), where $g_t(x):=|x|^3-\max(0,|x|-t)^3$.

$\endgroup$
  • $\begingroup$ The link to T^2 does not work because it goes through your library. $\endgroup$ – Bill Johnson Jun 3 '15 at 17:33
3
$\begingroup$
  1. Alas, the approach through the functions $g_t$ cannot possibly work. That is seen if one takes e.g. $n=2$, $a_1=4/5$, and $a_2=t=3/5$.

  2. Based on numerical evidence, the following approach promises to work. Without loss of generality (wlog) $n\ge2$ and $a_1\ge\dots\ge a_n\ge0$. Let $a:=a_1$, $b:=a_2$, and $Y:=|S-a_1\eta_1-a_2\eta_2|$. Then $$(1)\quad E Y^2=1-a^2-b^2$$ and $$(2)\quad EY^4=3\Big(\sum_3^n a_i^2\Big)^2-2\sum_3^n a_i^4\ge3(1-a^2-b^2)^2-2(1-a^2-b^2)\times\min(b^2,1-a^2-b^2), $$ since $\max_3^n a_i^2\le\min(b^2,1-a^2-b^2)$. By induction, it is enough to show that $$ (3)\quad E|S_2+Y|^p\le E\Big|\frac{S_2+T_2}{\sqrt2}+Y\Big|^p$$ for all $p\in(2,3)$ and nonnegative r.v.'s $Y$ subject to conditions (1) and (2), where $S_2:= \eta_1 a+\eta_2 b$ and $T_2$ is an independent copy of $S_2$. At that, by well-known results (see e.g. Hoeffding55), wlog the r.v. $Y$ takes at most $3$ values (say $0\le u\le v\le w$ with probabilities $r,s,1-r-s$).

Thus, the problem is reduced to a calculus problem on proving (say) the nonnegativity of a function of $8$ variables $p,a,b,u,v,w,r,s$ subject to a finite number of restrictions on the values of these variables. So, in principle this problem is solvable, but seems very involved computationally.

Addendum: Unfortunately, other numerical evidence shows that conditions (1) and (2) on $Y$ are not enough for (3) to hold in general. For instance, if $a = b = 11/21$, $p = 93/46$, $u = 0$, $v = 11/95$, $w = 71/61$, and $r\approx0.642$ and $s\approx0.025$ are such that $E Y^2=1-a^2-b^2$ and $EY^4=3(1-a^2-b^2)^2$, then the difference between the right-hand and left-hand sides of (3) is $\approx-0.000163$.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.