Let $(x\_1,...,x\_N)$ be points in $R^d$, and $\sigma=(\sigma\_1,...,\sigma\_N)$ are i.i.d. Rademacher variables (+1 or -1 with probability 0.5 each). (Or alternatively, $\sigma$ could be a standard Gaussian vector.) Also let $z$ in an infinite subset of $R^d$, and $\|\cdot\|\_2$ will denote the Euclidean norm. 
I am looking for a good upper estimate of:


$E\_{\sigma} \sup\_z \| \sum\_{n=1}^N \sigma\_n \frac{x\_n-z}{\|x\_n-z\|\_2} \|\_2$.


So far I only got the upper bound of $N$ (by triangle inequality), and the lower bound $\sqrt{N}$ by Jensen inequality, and I would like to know if there is a way to improve on the upper estimate or the lower estimate? 


If not, are there conditions (on the domain of $z$) under which the upper bound would reduce to ${\mathcal O}(\sqrt{N})$?


Edit: The points $x\_1,...,x\_N$ are assumed to be in general position (in a ball of radius $R$). 


Edit2: The domain of values of $z$ is an infinite set in $R^d$. Can be assumed to be bounded.


Edit3: The domain of $z$ is allowed to depend on $N$, $R$ (or on other assumed characteristics of the point set), but not on the individual points.


Edit4: I also want to avoid dependence on $d$.



Supremum of a stochastic process
First, if the subset $Z$ of $R^d$ of allowed values for $z$ is a **singleton**, then writing $e\_n=(x\_n-z)/\|x\_n-z\|\_2=(e\_{n,j})\_{j=1}^d$, we have
$$E\_\sigma \left\|\sum\sigma\_ne\_n\right\|\_2=E\_\sigma\sqrt{\sum\_j \left(\sum\_n\sigma\_ne\_{n,j}\right)^2}\le\sqrt{E\_\sigma\sum\_j\left(\sum\_n\sigma\_ne\_{n,j}\right)^2}$$
$$%=\sqrt{\sum\_jE\_\sigma\left[\left(\sum\_n\sigma\_ne\_{n,j}\right)^2\right]}
=\sqrt{\sum\_j\mathrm{Var}\_\sigma\left[\sum\_n\sigma\_ne\_{n,j}\right]}
=\sqrt{\sum\_j\sum\_n\mathrm{Var}\_\sigma\left[\sigma\_ne\_{n,j}\right]}$$
$$=\sqrt{\sum\_j\sum\_ne\_{n,j}^2}=\sqrt{\sum\_n \|e\_n\|\_2^2}=\sqrt{N}.$$
Second, **if $Z$ is allowed to depend on $N$**, then let $Z$ be of very small diameter $\delta>0$, relative to


* the distances $\|x\_i-x\_j\|\_2$,
* $N$, and
* the modulus of continuity of the function $z\mapsto \left\| \sum \sigma\_n e\_n\right\|\_2$,


then we should be able to get
for a fixed $z\_0\in Z$, and for all $\sigma$,
$$\sup\_{z\in Z}\left\|\sum\sigma\_ne\_n\right\|\_2 \le \left\|\sum\sigma\_ne\_n\text{ (for $z\_0$)}\right\|\_2 + \epsilon$$
and hence
$$E\_\sigma \left(\sup\_z\left\|\sum\sigma\_ne\_n\right\|\_2\right) \le E\_\sigma\left( \left\|\sum\sigma\_ne\_n\text{ (for $z\_0$)}\right\|\_2 + \epsilon\right) =
 \mathcal O(\sqrt{N}).$$


