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

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Statistical Inference, Econometric Analysis and Matrix Algebra

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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.

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Authors and Affiliations

  1. Fakultät für Mathematik, Technische Universität Dortmund, Dortmund, 44221, Germany

    Wilfried Hazod

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  1. Wilfried Hazod
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Correspondence to Wilfried Hazod .

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  1. TU Dresden Fakultät Wirtschaftswissenschaften, Professur für Quantitative Verfahren, insb. Ökonometrie, Mommsenstr. 12, Dresden, 01062, Germany

    Bernhard Schipp

  2. TU Dortmund Fakultät Statistik, Lehrstuhl für Wirtschaftsund Sozialstatistik, Vogelpothsweg 78, Dortmund, 44221, Germany

    Walter Kräer

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Hazod, W. (2009). Multiple Self-decomposable Laws on Vector Spaces and on Groups: The Existence of Background Driving Processes. In: Schipp, B., Kräer, W. (eds) Statistical Inference, Econometric Analysis and Matrix Algebra. Physica-Verlag HD. https://doi.org/10.1007/978-3-7908-2121-5_19

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