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VOL. 5 | 2009 A note on positive definite norm dependent functions
Alexander Koldobsky

Editor(s) Christian Houdré, Vladimir Koltchinskii, David M. Mason, Magda Peligrad


Let K be an origin symmetric star body in ℝn. We prove, under very mild conditions on the function f : [0, ∞)→ℝ, that if the function f(‖xK) is positive definite on ℝn, then the space (ℝn, ‖ ⋅ ‖K) embeds isometrically in L0. This generalizes the solution to Schoenberg’s problem and leads to progress in characterization of n-dimensional versions, i.e. random vectors X=(X1, …, Xn) in ℝn such that the random variables ∑aiXi are identically distributed for all a∈ℝn, up to a constant depending on ‖aK only.


Published: 1 January 2009
First available in Project Euclid: 2 February 2010

zbMATH: 1243.46018
MathSciNet: MR2797937

Digital Object Identifier: 10.1214/09-IMSCOLL502

Rights: Copyright © 2009, Institute of Mathematical Statistics


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