Open Access
April, 1992 Operator Exponents of Probability Measures and Lie Semigroups
Zbigniew J. Jurek
Ann. Probab. 20(2): 1053-1062 (April, 1992). DOI: 10.1214/aop/1176989817


A notion of $U$-exponents of a probability measure on a linear space is introduced. These are bounded linear operators and it is shown that the set of all $U$-exponents forms 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 in question. Then 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.


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Zbigniew J. Jurek. "Operator Exponents of Probability Measures and Lie Semigroups." Ann. Probab. 20 (2) 1053 - 1062, April, 1992.


Published: April, 1992
First available in Project Euclid: 19 April 2007

zbMATH: 0779.60007
MathSciNet: MR1159585
Digital Object Identifier: 10.1214/aop/1176989817

Primary: 60B12
Secondary: 20M20 , 22A15

Keywords: $U$-exponent , Decomposability semigroup , Haar measure , Lie wedge , operator-self-decomposable measure , Schur lemma , self-decomposable measure , tangent space , Wedge

Rights: Copyright © 1992 Institute of Mathematical Statistics

Vol.20 • No. 2 • April, 1992
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