Multiple Self-decomposable Laws on Vector Spaces and on Groups: The Existence of Background Driving Processes

Wilfried Hazod ()
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Wilfried Hazod: Technische Universität Dortmund, Fakultät für Mathematik

A chapter in Statistical Inference, Econometric Analysis and Matrix Algebra, 2009, pp 295-307 from Springer

Abstract: Abstract Following K. Urbanik, we define for simply connected nilpotent Lie groups G multiple self-decomposable laws as follows: For a fixed continuous one-parameter group (T t ) of automorphisms put $$L^{(0)} : = M^1 \left( G \right)\,and\,L^{(m + 1)} : = \{ \mu \in M^1 \left( G \right):\forall t > 0\ \exists\ v(t) \in L^{(m)} :\mu = T_t (\mu )*v(t)\} \,for \,m \ge 0.$$ Under suitable commutativity assumptions it is shown that also for m > 0 there exists a background driving Lévy process with corresponding continuous convolution semigroup (v s)s≥0 determining μ and vice versa. Precisely, μ and v s are related by iterated Lie Trotter formulae.

Keywords: Convolution Semigroup; Multivariate Anal; Weak Operator Topology; Convolution Structure; Logarithmic Moment (search for similar items in EconPapers)
Date: 2009
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-7908-2121-5_19

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DOI: 10.1007/978-3-7908-2121-5_19

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