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Comparison and Deviation from a Representation Formula

  • Christian Houdré
Part of the Trends in Mathematics book series (TM)

Abstract

Let X ~ ID(b, ΣE, v), i.e., let X be a d-dimensional infinitely divisible random vector with characteristic function
$$\varphi (t) = \exp \left\{ {i\langle t,b\rangle - \frac{1}{2}\langle \sum t ,t\rangle + \int_{{\mathbb{R}^d}} {({e^{i\langle t,u\rangle }} - 1 - i\langle t,u\rangle 1(\left| u \right|} < 1))\nu (du)} \right\},$$
(1.1)
where t, b ∈ ℝ d , Σ is a positive semidefinite d × d matrix and v (the Lévy measure) is a positive measure on B(ℝ d ), the Borel σ-algebra of ℝ d , without atom at the origin and such that \(\int_{\mathbb{R}^d } {(\left| u \right|} ^2 \wedge 1)\nu (du) < + \infty (\langle \cdot , \cdot \rangle {e_{1}}, \ldots ,{e_{n}},e_{i}^{2} = - 1, \) and ∣ · ∣ are respectively the Euclidean inner product and norm in ℝ d ).

Keywords

Lipschitz Function Isoperimetric Inequality Representation Formula Positive Definite Function Deviation Inequality 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science+Business Media New York 1998

Authors and Affiliations

  • Christian Houdré
    • 1
  1. 1.Southeast Applied Analysis CenterSchool of Mathematics Georgia Institute of TechnologyAtlantaUSA

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