This relate to that paper: 


<http://www.stat.purdue.edu/docs/research/tech-reports/1982/tr82-17.pdf>


Let $U\_1,...,Un$ be iid uniform on (0,1).
Set $L\_n=\max\_{i\leq n} U\_i$. 


Also $S(n)= \inf\{i\leq n| U\_i = L\_n \}$ the time were the highest value is attained and


$Z(n)= \inf\{i\leq n| U\_i = L\_{S(n)-1} \} \vee 0$ the time were the highest value before the very highest is attained (not necessary the second highest globally!)


Proof that there are $V$ ,$V' \sim Exp(1)$ and $W$, $W' \sim U(0,1)$ all mutually independent, such that:


$(n(1-L\_n),(S(n)-1)(1-\frac{L\_{S(n)-1}}{L\_n}),\frac{S(n)}{n},\frac{Z(n)}{S(n)-1})\overset{d}{\rightarrow}(V,V',W,W').$ 


A proof sketch will be sufficient.


I also posted the question here:<https://math.stackexchange.com/questions/1844238/limit-of-a-probability-vector>



Question on a random vector
For $1 \leq s' < s \leq n$ and $t,t' \in \mathbb{R}$, one has
$$
n(1-L\_n) > t, (S(n)-1)(1-\frac{L\_{S(n)-1}}{L\_n}) > t',S(n)=s, Z(n) = s'
$$
precisely when 
$$
U\_1,\dots,U\_{s'-1} < U\_{s'},\\
U\_{s'+1},\dots,U\_{s-1} \leq U\_{s'}, \\
U\_{s+1},\dots,U\_{n} \leq U\_{s},\\
U\_{s'} < \left(1 - \frac{t'\_+}{s-1} \right)\_+ U\_s, \\
 U\_{s} < \left(1 - \frac{t\_+}{n} \right)\_+ ,
$$
where $x\_+ = \max(x,0)$. Setting $\alpha =\left(1 - \frac{t\_+}{n} \right)\_+ $ and $\alpha' = \left(1 - \frac{t'\_+}{s-1} \right)\_+$, the probability of these events is 
$$
\mathbb{E}[U\_{s'}^{s-2} \mathbb{1}\_{U\_{s'} \leq \alpha' U\_s} U\_{s}^{n-s} \mathbb{1}\_{U\_{s} \leq \alpha} ] = \mathbb{E}[\frac{(\alpha' U\_s)^{s-1}}{s-1}U\_{s}^{n-s} \mathbb{1}\_{U\_{s} \leq \alpha}] \\
= \frac{\alpha'^{s-1} \alpha^n}{n(s-1)} = \frac{1}{n(s-1)}\left(1 - \frac{t'\_+}{s-1} \right)\_+^{s-1} \left(1 - \frac{t\_+}{n} \right)\_+^n
$$
The conclusion follows from this computation and from the following fact : for any $w,w' \in ]0,1]$, one has
$$
\sum\_{\substack{2\leq s \leq wn \\ 1 \leq s' \leq w'(s-1)}} \frac{1}{n(s-1)}\left(1 - \frac{t'\_+}{s-1} \right)\_+^{s-1} \left(1 - \frac{t\_+}{n} \right)\_+^n \longrightarrow w w' e^{-t'\_+ - t\_+} ,
$$
as $n \rightarrow + \infty$.


