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Regularizing mappings of Lévy measures

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  • Barndorff-Nielsen, Ole E.
  • Thorbjørnsen, Steen

Abstract

In this paper we introduce and study a regularizing one-to-one mapping from the class of one-dimensional Lévy measures into itself. This mapping appeared implicitly in our previous paper [O.E. Barndorff-Nielsen, S. Thorbjørnsen, A connection between free and classical infinite divisibility, Inf. Dim. Anal. Quant. Probab. 7 (2004) 573-590], where we introduced a one-to-one mapping [Upsilon] from the class of one-dimensional infinitely divisible probability measures into itself. Based on the investigation of in the present paper, we deduce further properties of [Upsilon] . In particular it is proved that [Upsilon] maps the class of selfdecomposable laws onto the so called Thorin class . Further, partly motivated by our previous studies of infinite divisibility in free probability, we introduce a one-parameter family of one-to-one mappings , which interpolates smoothly between [Upsilon] ([alpha]=0) and the identity mapping on ([alpha]=1). We prove that each of the mappings shares many of the properties of [Upsilon]. In particular, they are representable in terms of stochastic integrals with respect to associated Lévy processes.

Suggested Citation

  • Barndorff-Nielsen, Ole E. & Thorbjørnsen, Steen, 2006. "Regularizing mappings of Lévy measures," Stochastic Processes and their Applications, Elsevier, vol. 116(3), pages 423-446, March.
    • Handle: RePEc:eee:spapps:v:116:y:2006:i:3:p:423-446
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    Cited by:

    1. Barndorff-Nielsen, Ole E. & Maejima, Makoto, 2008. "Semigroups of Upsilon transformations," Stochastic Processes and their Applications, Elsevier, vol. 118(12), pages 2334-2343, December.

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