Asymptotic radius of the smallest enclosing ball 
*

*Let $X_1,..., X_n$ be i.i.d. $d$-dimensional standard  normal random variables, and let $R_n$ be the radius of the smallest ball containing $X_1,...,X_n$. What is known about the distribution of $R_n$ as $n \to \infty$?

*Are there some examples for the asymptotics of $R_n$ when the independence assumption does not hold? I would appreciate any reference or suggestion here, also for a non-Gaussian case.
 A: This  is to confirm and give more details to Anthony Quas' comment.
Let's  start with a bit of wishful thinking.   Denote by $P_n$ the convex hul of the points $X_1,\dotsc, X_n$.$\newcommand{\bR}{\mathbb{R}}$ $\newcommand{\bE}{\mathbb{E}}$ Denote by $B_R(p)$  the ball or radius $R$ centered at $p\in\bR^d$.Suppose that for any $n\geq 3$ there exists  radii $R_n^-<R_n^+$ such that.


*

*$B_{R_n^-}(0)\subset P_n \subset B_{R_n^+}(0)$ a.s. for large $n$.

*$R_n^-\sim R_n^+$ as $n\to \infty$, i.e.,


$$ \lim_{n\to\infty} \frac{R_n^+}{R_n^-}=1. $$
Then, almost surely, 
$$\frac{R_n}{R_n^-}\to  1. $$
Fortunately  such radii  can be found and they have the growth rate suggested  by Anthony Quas.  This is all contained  in a preprint of Fresen & Vitale  arXiv: 1402.2718
Let me give a few more details.  Denote by $\mu$ The standard Gaussian measure on $\bR^d$. For each  $\delta>0$ we denote by $F_\delta$ the intersection of all closed half-spaces $S\subset \bR^d$ such that $\mu(S)\geq 1-\delta$.
The  floating body is then the convex set $F_{1/n}$. Due to the radial symmetry of $\mu$ you can see that  $F_{1/n}$  is a ball of radius $r_n$ centered at the orgin,where $r_n$ is found  from the equality
$$\Psi(r_n)=\frac{1}{n}, \;\;\Psi(x):=\frac{1}{\sqrt{2\pi}}\int_{x}^\infty e^{-t^2/2} dt=\frac{1}{n}. $$
From the known estimates
$$ \Big(\frac{1}{x}-\frac{1}{x^3}\Big)\gamma(x)  < \Psi(x) <\frac{1}{x}\gamma(x),\;\;\;\;\gamma(x)=\frac{1}{\sqrt{2\pi}} e^{-x^2/2}. $$
We deduce that
$$r_n=\sqrt{2\log n} +o(1),  \;\;\mbox{as}\; n\to\infty. $$
In the above preprint construct  (explicit) sequences of positive real numbers $\newcommand{\ve}{\varepsilon}$ $\ve_1(n),\ve_2(n)$ converging to $0$ with the following property:  with probability $1$, there  exists  a natural number  $N$ such that for all $n>N$
$$ \big(1-\ve_1(n)\;\big) F_{1/n}\subset P_n\subset  \big(1+\ve_2(n)\;\big) F_{1/n}. $$
Thus proves that 
$$ \frac{R_n}{\sqrt{2\log n}}\to 1\;\; \mbox{a.s.}. $$
