## The Annals of Probability

- Ann. Probab.
- Volume 7, Number 3 (1979), 494-499.

### Infinitely Divisible Distributions with Unimodal Levy Spectral Functions

#### Abstract

The class of infinitely divisible characteristic functions which have unimodal Levy spectral functions is determined. It is shown that membership in this class is related to solutions of the equations $\phi(u) = \phi^r(ru)\phi_r(u)$, where $r \in (0, 1)$ and $\phi$ and $\phi_r$ are characteristic functions. We point out how elements of this class can serve as limit laws as well as some connections between this class and the class of self-decomposable characteristic functions.

#### Article information

**Source**

Ann. Probab. Volume 7, Number 3 (1979), 494-499.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176995049

**Digital Object Identifier**

doi:10.1214/aop/1176995049

**Mathematical Reviews number (MathSciNet)**

MR528326

**Zentralblatt MATH identifier**

0401.60015

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60E05: Distributions: general theory

Secondary: 60F05: Central limit and other weak theorems

**Keywords**

Infinitely divisible characteristic function unimodal Levy spectral function self-decomposable characteristic function u.a.n. system of random variables central limit theorem

#### Citation

O'Connor, Thomas A. Infinitely Divisible Distributions with Unimodal Levy Spectral Functions. Ann. Probab. 7 (1979), no. 3, 494--499. doi:10.1214/aop/1176995049. https://projecteuclid.org/euclid.aop/1176995049