## The Annals of Probability

### Operator Exponents of Probability Measures and Lie Semigroups

Zbigniew J. Jurek

#### Abstract

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.

#### Article information

Source
Ann. Probab., Volume 20, Number 2 (1992), 1053-1062.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176989817

Digital Object Identifier
doi:10.1214/aop/1176989817

Mathematical Reviews number (MathSciNet)
MR1159585

Zentralblatt MATH identifier
0779.60007

JSTOR