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VOL. 1 | 2018 Chapter 23. A Simple Round-up on the Validity of a.s Convergence after Partial Modification of the Probability Law and Application
Gane Samb LO

Editor(s) Hamet SEYDI, Gane Samb LO, Aboubakary DIAKHABY

## 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

Published: 1 January 2018
First available in Project Euclid: 26 September 2019

Digital Object Identifier: 10.16929/sbs/2018.100-04-06

Subjects:
Primary: 60F15 , 60Fxx

Keywords: convergence in almost sure , Convergence in distribution , convergence in probability , order statistics of uniform random variables , partial change of margins , partial sums of standard exponential , preservation of type of limits , types of convergence of random variables