The 123 theorem of Probability Theory and Copositive Matrices

Abstract

Alon and Yuster give for independent identically distributed real or vector valued random variablesX, Y combinatorially proved estimates of the form Prob(∥X − Y∥ ≤ b) ≤ c Prob(∥X − Y∥ ≤ a). We derivethese using copositive matrices instead. By the same method we also give estimates for the real valued case,involving X + Y and X − Y, due to Siegmund-Schultze and von Weizsäcker as generalized by Dong, Li andLi. Furthermore, we formulate a version of the above inequalities as an integral inequality for monotonefunctions.

Keywords

• probabilistic inequalities
• copositivity
• integral inequality
• 60e15
• 15b48
• 26d10