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Operator exponents of probability measures and Lie semigroups. (English) Zbl 0779.60007

\(U\)-exponents of probability measure on a linear space are introduced. These are bounded linear operators, and it is shown that the set of all \(U\)-exponents form a Lie wedge for full measures on finite-dimensional spaces. This allows the construction of \(U\)-exponents commuting with the symmetry group of a measure. The set of all commuting exponents is described and elliptically symmetric measures are characterized in terms of their Fourier transforms. Also, self-decomposable measures are identified among those which are operator-self-decomposable. Finally, \(S\)-exponents of infinitely divisible measures are discussed.
Reviewer: Y.Asoo (Okayama)

MSC:

60B15 Probability measures on groups or semigroups, Fourier transforms, factorization
22A15 Structure of topological semigroups
20M20 Semigroups of transformations, relations, partitions, etc.
60E07 Infinitely divisible distributions; stable distributions
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