## Statistics and Probability African Society Editions

- SPAS Books Series
- Volume 1, 2018, 437-465

### 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.

#### Chapter information

**Source***A Collection of Papers in Mathematics and Related Sciences, a Festschrift in Honour of the Late Galaye Dia*, (Calgary, Alberta, 2018)

**Dates**

First available in Project Euclid: 26 September 2019

**Permanent link to this document**

https://projecteuclid.org/euclid.spaseds/1569509483

**Digital Object Identifier**

doi:10.16929/sbs/2018.100-04-06

**Subjects**

Primary: 60F15: Strong theorems 60FXX

**Keywords**

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

#### Citation

LO, Gane Samb. Chapter 23. A Simple Round-up on the Validity of a.s Convergence after Partial Modification of the Probability Law and Application. A Collection of Papers in Mathematics and Related Sciences, 437--465, Statistics and Probability African Society, Calgary, Alberta, 2018. doi:10.16929/sbs/2018.100-04-06. https://projecteuclid.org/euclid.spaseds/1569509483