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Regularizing mappings of Lévy measures
- Source :
- Barndorff-Nielsen, O E & Thorbjørnsen, S 2006, ' Regularizing mappings of Levy measures ', Stochastic Processes and Their Applications, vol. 116, pp. 423-446 . https://doi.org/10.1016/j.spa.2005.09.011, Barndorff-Nielsen, O E & Thorbjørnsen, S 2006, ' Regularizing mappings of Lévy measures ', Stochastic Processes and Their Applications, vol. 116, no. 3, pp. 423-446 . https://doi.org/10.1016/j.spa.2005.09.011
- Publication Year :
- 2006
- Publisher :
- Elsevier BV, 2006.
-
Abstract
- In this paper we introduce and study a regularizing one-to-one mapping ϒ 0 from the class of one-dimensional Levy measures into itself. This mapping appeared implicitly in our previous paper [O.E. Barndorff-Nielsen, S. Thorbjornsen, 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 ϒ from the class ID ( * ) of one-dimensional infinitely divisible probability measures into itself. Based on the investigation of ϒ 0 in the present paper, we deduce further properties of ϒ . In particular it is proved that ϒ maps the class L ( * ) of selfdecomposable laws onto the so called Thorin class T ( * ) . Further, partly motivated by our previous studies of infinite divisibility in free probability, we introduce a one-parameter family ( ϒ α ) α ∈ [ 0 , 1 ] of one-to-one mappings ϒ α : ID ( * ) → ID ( * ) , which interpolates smoothly between ϒ ( α = 0 ) and the identity mapping on ID ( * ) ( α = 1 ). We prove that each of the mappings ϒ α shares many of the properties of ϒ . In particular, they are representable in terms of stochastic integrals with respect to associated Levy processes.
- Subjects :
- Statistics and Probability
Pure mathematics
Class (set theory)
Free probability
Selfdecomposability
Monotonic function
Lévy process
Mittag–Leffler law
symbols.namesake
Modelling and Simulation
Mittag-Leffler function
Infinite divisibility
Thorin class
Mathematics
Probability measure
Levy processes
Completely monotone function
Applied Mathematics
Mathematical analysis
Bondesson class
Modeling and Simulation
symbols
Probability distribution
Mittag–Leffler function
Subjects
Details
- ISSN :
- 03044149
- Volume :
- 116
- Database :
- OpenAIRE
- Journal :
- Stochastic Processes and their Applications
- Accession number :
- edsair.doi.dedup.....58f59e318b295befbf324470c9f178f1