Operator exponents of probability measures and Lie semigroups
\(U\)-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.
scientific article; zbMATH DE number 89801