Open Access
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
Chapter Author(s) Gane Samb LO
Editor(s) Hamet SEYDI, Gane Samb LO, Aboubakary DIAKHABY

Abstract

Let U1, U2, ... be a sequence of independent and uniformly distributed random variables on (0,1) defined on the same probability space. Let U1,n...Un,n be the order statistics of the sample U1, U2,...Un of size n1. Let (k(n))n1 be a sequence of integers such that 1k(n)n and k(n)+. We prove that nUk(n),n/k(n)1 a.s as n+. 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

Rights: Copyright © 2018 The Statistics and Probability African Society

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