# The 123 theorem of Probability Theory and Copositive Matrices

Special Matrices (2014)

• Volume: 2, Issue: 1, page 155-164, electronic only
• ISSN: 2300-7451

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## Abstract

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Alon and Yuster give for independent identically distributed real or vector valued random variables X, Y combinatorially proved estimates of the form Prob(∥X − Y∥ ≤ b) ≤ c Prob(∥X − Y∥ ≤ a). We derive these 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 and Li. Furthermore, we formulate a version of the above inequalities as an integral inequality for monotone functions.

## How to cite

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Alexander Kovačec, Miguel M. R. Moreira, and David P. Martins. "The 123 theorem of Probability Theory and Copositive Matrices." Special Matrices 2.1 (2014): 155-164, electronic only. <http://eudml.org/doc/267310>.

@article{AlexanderKovačec2014,
abstract = {Alon and Yuster give for independent identically distributed real or vector valued random variables X, Y combinatorially proved estimates of the form Prob(∥X − Y∥ ≤ b) ≤ c Prob(∥X − Y∥ ≤ a). We derive these 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 and Li. Furthermore, we formulate a version of the above inequalities as an integral inequality for monotone functions.},
author = {Alexander Kovačec, Miguel M. R. Moreira, David P. Martins},
journal = {Special Matrices},
keywords = {probabilistic inequalities; copositivity; integral inequality},
language = {eng},
number = {1},
pages = {155-164, electronic only},
title = {The 123 theorem of Probability Theory and Copositive Matrices},
url = {http://eudml.org/doc/267310},
volume = {2},
year = {2014},
}

TY - JOUR
AU - Alexander Kovačec
AU - Miguel M. R. Moreira
AU - David P. Martins
TI - The 123 theorem of Probability Theory and Copositive Matrices
JO - Special Matrices
PY - 2014
VL - 2
IS - 1
SP - 155
EP - 164, electronic only
AB - Alon and Yuster give for independent identically distributed real or vector valued random variables X, Y combinatorially proved estimates of the form Prob(∥X − Y∥ ≤ b) ≤ c Prob(∥X − Y∥ ≤ a). We derive these 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 and Li. Furthermore, we formulate a version of the above inequalities as an integral inequality for monotone functions.
LA - eng
KW - probabilistic inequalities; copositivity; integral inequality
UR - http://eudml.org/doc/267310
ER -

## References

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1. [1] H. Alzer, Private communication.
2. [2] N. Alon and R. Yuster, The 123 Theorem and Its Extensions, J. of Combin. Theory, Ser. A 72, 321-331 (1995). Zbl0834.60015
3. [3] H. Bauer, Probability theory and elements of measure theory, Academic Press, 1981. Zbl0466.60001
4. [4] R.P. Boas, A Primer of Real Functions, 3rd Edition, MAA, 1981.
5. [5] R.W. Cottle, C.E. Habetler and G.J. Lemke, On classes of copositive matrices, Linear Algebra Appl. 3, 295-310 (1970). [WoS] Zbl0196.05602
6. [6] Z. Dong, J. Li and W.V. Li, A Note on Distribution-Free Symmetrization Inequalities, J. Theor. Probab. 2014 (DOI 10.1007/s10959-014-0538-z) [Crossref] Zbl06516680
7. [7] M. Loève, Probability Theory, I, 4th Edition, Springer 1977.
8. [8] D.H. Martin, Finite criteria for conditional definiteness of quadratic forms, Linear Algebra Appl. 39, 9-21 (1981). Zbl0464.15012
9. [9] R. Siegmund-Schultze and H. von Weizsäcker, Level crossing probabilities I: One-dimensional random walks and symmetrization, Adv. Math. 208, 672-679 (2007).[WoS] Zbl1108.60040

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