Need an upper bound for a simple expectation involving Rademacher random variables.

Question

I need an upper bound on the following expectation:
$$E_{x_1,\dots,x_n} \left| \sum_{i=1}^n a_i x_i \right| \quad,\quad \mbox{s.t.} \sum_{i=1}^n a_i^2=1.$$

Here $x_i$'s are iid Rademacher random variables ($x_i=+1$ or $x_i=-1$, each with probability $1/2$) and |.| is the absolute value. The real vector $(a_1,…a_n)$, as shown, is on the unit sphere.

What is the best upper bound you know for this expectation? Getting a bound using Holder's inequality is straightforward,
\begin{equation}
E_{x_1,\dots,x_n} \left| \sum_{i=1}^n a_i x_i \right| \leq E_{x_1,\dots,x_n} \|\boldsymbol{a}\|_1 \|\boldsymbol{x}\|_{\infty} = \|\boldsymbol{a}\|_1
\end{equation}
However, I am hoping for a tighter bound.

Thanks!

Answer 1

using $E\big[X\big]^2 \leq E\big[X^2\big]$ (Jensen) allows you to (i) linearize the absolute value and (ii) use explicitly use $\sum_{i=1}^n a_i^2=1$.

$\Big(E_{\mathbf x} \big[ \vert\sum_{i=1}^n a_i X_i \vert\big]\Big)^2$
$\leq E_{\mathbf x} \big[ (\sum_{i=1}^n a_i X_i \big)^2\big]$
$=E_{\mathbf x} \big[ \sum_{i=1}^n a_i^2 X_i^2 \big] + E_{\mathbf x} \big[ \sum_{i=1}^n\sum_{j\neq i} a_ia_j X_iX_j \big]$
$= \sum_{i=1}^n a_i^2 E_{\mathbf x} \big[X_i^2 \big] + \sum_{i=1}^n\sum_{j\neq i} a_ia_j E_{\mathbf x} \big[X_i\big]E_{\mathbf x} \big[X_j \big]$
$= \sum_{i=1}^n a_i^2 +0$
$=1$

taking square roots
$E_{\mathbf x} \big[ \vert\sum_{i=1}^n a_i X_i \vert\big]\leq 1$

recall $\big\Vert \mathbf a\big\Vert_2 \leq \big\Vert \mathbf a\big\Vert_1$
by triangle inequality so this is a sharper bound.