A Probabilistic Inequality Related to Negative Definite Functions
2013; Springer International Publishing; Linguagem: Inglês
10.1007/978-3-0348-0490-5_5
ISSN2297-0428
AutoresMikhail Lifshits, René L. Schilling, I. S. Tyurin,
Tópico(s)Functional Equations Stability Results
ResumoWe 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.
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