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The 123 theorem of Probability Theory and Copositive Matrices

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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.
  • Department of Mathematics, University of Coimbra, 3001-501, Coimbra, Portugal
  • Rua Luís de Camões, Nr. 102, 1300-360, Lisboa, Portugal
  • Rua D. Manuel I, Edif. Império Porta 2-2D, 5370-412 Mirandela, Portugal
  • [1] H. Alzer, Private communication.
  • [2] N. Alon and R. Yuster, The 123 Theorem and Its Extensions, J. of Combin. Theory, Ser. A 72, 321-331 (1995).
  • [3] H. Bauer, Probability theory and elements of measure theory, Academic Press, 1981.
  • [4] R.P. Boas, A Primer of Real Functions, 3rd Edition, MAA, 1981.
  • [5] R.W. Cottle, C.E. Habetler and G.J. Lemke, On classes of copositive matrices, Linear Algebra Appl. 3, 295-310 (1970). [WoS]
  • [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]
  • [7] M. Loève, Probability Theory, I, 4th Edition, Springer 1977.
  • [8] D.H. Martin, Finite criteria for conditional definiteness of quadratic forms, Linear Algebra Appl. 39, 9-21 (1981).
  • [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]
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