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:http://math.stackexchange.com/questions/1844238/limit-of-a-probability-vector

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