Abstract
Let $U_{1}$, $U_{2}$, ... be a sequence of independent and uniformly distributed random variables on $(0,1)$ defined on the same probability space. Let $U_{1,n} \le ...\le U_{n,n}$ be the order statistics of the sample $U_{1}$, $U_{2}$,...$U_{n}$ of size $n \geq 1$. Let $(k(n))_{n\geq 1}$ be a sequence of integers such that $1\leq k(n) \leq n$ and $k(n) \longrightarrow +\infty$. We prove that $nU_{k(n),n}/k(n) \longrightarrow 1$ a.s as $n \longrightarrow +\infty$. We take the opportunity to make a simple Round-up on the validity of different type of convergences when the sequence of random variables is replaced by another sequence preserving parts of the probability law of the original sequence.
Information
Digital Object Identifier: 10.16929/sbs/2018.100-04-06