## The Annals of Probability

- Ann. Probab.
- Volume 3, Number 1 (1975), 70-79.

### Equivalence of Infinitely Divisible Distributions

William N. Hudson and Howard G. Tucker

#### Abstract

If $F$ is an infinitely divisible distribution function without a Gaussian component whose Levy spectral measure $M$ is absolutely continuous and $M(\mathbb{R}^1\backslash\{0\}) = \infty$, then $F$ is shown to have an a.e. positive density over its support; this support of $F$ is always an interval of the form $(-\infty, \infty), (-\infty, a\rbrack$ or $\lbrack a, \infty)$. In addition, sufficient conditions are obtained for two infinitely divisible distribution functions without Gaussian components to be absolutely continuous with respect to each other, i.e., equivalent.

#### Article information

**Source**

Ann. Probab., Volume 3, Number 1 (1975), 70-79.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176996449

**Digital Object Identifier**

doi:10.1214/aop/1176996449

**Mathematical Reviews number (MathSciNet)**

MR372944

**Zentralblatt MATH identifier**

0303.60011

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60E05: Distributions: general theory

**Keywords**

Infitely divisible distribution functions and characteristic functions absolute continuity of measures equivalence of measures

#### Citation

Hudson, William N.; Tucker, Howard G. Equivalence of Infinitely Divisible Distributions. Ann. Probab. 3 (1975), no. 1, 70--79. doi:10.1214/aop/1176996449. https://projecteuclid.org/euclid.aop/1176996449