Chapter 23. A Simple Round-up on the Validity of a.s Convergence after Partial Modification of the Probability Law and Application

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.