Home

About

Advanced Search

Browse by Discipline

Scientific Societies

E-print Alerts

Add E-prints

E-print Network
FAQHELPSITE MAPCONTACT US


  Advanced Search  

 
The 123 Theorem and its extensions and Raphael Yuster
 

Summary: The 123 Theorem and its extensions
Noga Alon
and Raphael Yuster
Department of Mathematics
Raymond and Beverly Sackler Faculty of Exact Sciences
Tel Aviv University, Tel Aviv, Israel
Abstract
It is shown that for every b > a > 0 and for every two independent identically distributed
real random variables X and Y
Prob[|X - Y | b] < (2 b/a - 1)Prob[|X - Y | a].
This is tight for all admissible pairs a, b. Higher dimensional extensions are also considered.
1 Introduction
Our first result in this note is the following theorem, which we name after the three constants in
its statement.
Theorem 1.1 (The 123 Theorem) Let X and Y be two independent, identically distributed real
random variables. Then
Prob[|X - Y | 2] < 3Prob[|X - Y | 1].
The problem of determining the smallest possible constant C so that for every two independent,
identically distributed (=i.i.d.) real random variables the inequality
Prob[|X - Y | 2] CProb[|X - Y | 1]

  

Source: Alon, Noga - School of Mathematical Sciences, Tel Aviv University

 

Collections: Mathematics