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

In: Statistical Inference, Econometric Analysis and Matrix Algebra

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  • Wilfried Hazod

    (Technische Universität Dortmund, Fakultät für Mathematik)

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.

Suggested Citation

  • Wilfried Hazod, 2009. "Multiple Self-decomposable Laws on Vector Spaces and on Groups: The Existence of Background Driving Processes," Springer Books, in: Bernhard Schipp & Walter Kräer (ed.), Statistical Inference, Econometric Analysis and Matrix Algebra, pages 295-307, Springer.
    • Handle: RePEc:spr:sprchp:978-3-7908-2121-5_19
    DOI: 10.1007/978-3-7908-2121-5_19
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