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Published in:
2009 | OriginalPaper | Chapter
Multiple Self-decomposable Laws on Vector Spaces and on Groups: The Existence of Background Driving Processes
Author : Wilfried Hazod
Published in: Statistical Inference, Econometric Analysis and Matrix Algebra
Publisher: Physica-Verlag HD
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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.