## The Annals of Probability

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

### Operator Exponents of Probability Measures and Lie Semigroups

#### 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**

links.jstor.org

**Subjects**

Primary: 60B12: Limit theorems for vector-valued random variables (infinite- dimensional case)

Secondary: 22A15: Structure of topological semigroups 20M20: Semigroups of transformations, etc. [See also 47D03, 47H20, 54H15]

**Keywords**

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

#### Citation

Jurek, Zbigniew J. Operator Exponents of Probability Measures and Lie Semigroups. Ann. Probab. 20 (1992), no. 2, 1053--1062. doi:10.1214/aop/1176989817. https://projecteuclid.org/euclid.aop/1176989817