Jurek, Zbigniew J. Operator exponents of probability measures and Lie semigroups. (English) Zbl 0779.60007 Ann. Probab. 20, No. 2, 1053-1062 (1992). \(U\)-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) Cited in 1 ReviewCited in 2 Documents 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 Keywords:\(U\)-exponents; symmetry group of a measure; self-decomposable measures; operator-self-decomposable; infinitely divisible measures × Cite Format Result Cite Review PDF Full Text: DOI