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A Probabilistic Inequality Related to Negative Definite Functions

  • Mikhail LifshitsEmail author
  • René L. Schilling
  • Ilya Tyurin
Conference paper
Part of the Progress in Probability book series (PRPR, volume 66)

Abstract

We prove that for any pair of i.i.d. random vectors X,Y in \(\mathbb{R}^n\) and any real-valued continuous negative definite function \(\psi\; : \;\mathbb{R}^n\rightarrow\mathbb{R}\) the inequality
$$\mathbb{E}\;\psi\;(X\;-\;Y)\leqslant\mathbb{E}\;\psi\;(X\;+\;Y).$$
holds. In particular, for \(\alpha\;\in\;(0,2]\) and the Euclidean norm \(\|\cdot\|_2\) one has
$$\mathbb{E}\|(X\;-\;Y)\|^\alpha_2\leqslant\mathbb{E}\|(X\;+\;Y)\|^\alpha_2.$$
The latter inequality is due to A. Buja et al. [4] where it is used for some applications in multivariate statistics. We show a surprising connection with bifractional Brownian motion and provide some related counter-examples.

Keywords

Bifractional Brownian motion moment inequalities Bernstein functions negative definite functions 

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Copyright information

© Springer Basel 2013

Authors and Affiliations

  • Mikhail Lifshits
    • 1
    Email author
  • René L. Schilling
    • 2
  • Ilya Tyurin
    • 3
  1. 1.Department of Mathematics and MechanicsSt. Petersburg State UniversityStary PeterhofRussia
  2. 2.Institute of Mathematical StochasticsTU DresdenDresdenGermany
  3. 3.Department of Mechanics and MathematicsMoscow State UniversityMoscowRussia

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