Summary: The 123 Theorem and its extensions
and Raphael Yuster
Department of Mathematics
Raymond and Beverly Sackler Faculty of Exact Sciences
Tel Aviv University, Tel Aviv, Israel
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.
Our first result in this note is the following theorem, which we name after the three constants in
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]