Regularizing mappings of Lévy measures

Ole Barndorff-Nielsen and Steen Thorbjørnsen

Stochastic Processes and their Applications, 2006, vol. 116, issue 3, 423-446

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.

Keywords: Bondesson; class; Completely; monotone; function; Free; probability; Infinite; divisibility; Lévy; processes; Mittag-Leffler; law; Mittag-Leffler; function; Selfdecomposability; Thorin; class (search for similar items in EconPapers)
Date: 2006
References: View complete reference list from CitEc
Citations: View citations in EconPapers (1)

Downloads: (external link)
http://www.sciencedirect.com/science/article/pii/S0304-4149(05)00140-7
Full text for ScienceDirect subscribers only

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:eee:spapps:v:116:y:2006:i:3:p:423-446

Ordering information: This journal article can be ordered from
http://http://www.elsevier.com/wps/find/supportfaq.cws_home/regional
https://shop.elsevie ... _01_ooc_1&version=01

Access Statistics for this article

Stochastic Processes and their Applications is currently edited by T. Mikosch

More articles in Stochastic Processes and their Applications from Elsevier
Bibliographic data for series maintained by Catherine Liu ().