## The Annals of Probability

### Nested classes of $C$-decomposable laws

John Bunge

#### Abstract

A random variable $X$ is C-decomposable if $X =_D cX + Y_c$ for all $c$ in $C$, where $_c$ is a random variable independent of $X$ and $C$ is a closed multiplicative subsemigroup of [0, 1]. $X$ is self-decomposable if $C = [0, 1]$ . Extending an idea of Urbanik in the self-decomposable case, we define a decreasing sequence of subclasses of the class of $C$-decomposable laws, for any $C$. We give a structural representation for laws in these classes, and we show that laws in the limiting subclass are infinitely divisible. We also construct noninfinitely divisible examples, some of which are continuous singular.

#### Article information

Source
Ann. Probab., Volume 25, Number 1 (1997), 215-229.

Dates
First available in Project Euclid: 18 June 2002

https://projecteuclid.org/euclid.aop/1024404286

Digital Object Identifier
doi:10.1214/aop/1024404286

Mathematical Reviews number (MathSciNet)
MR1428507

Zentralblatt MATH identifier
0873.60005

Subjects
Primary: 60E05: Distributions: general theory
Secondary: 60F05: Central limit and other weak theorems

#### Citation

Bunge, John. Nested classes of $C$-decomposable laws. Ann. Probab. 25 (1997), no. 1, 215--229. doi:10.1214/aop/1024404286. https://projecteuclid.org/euclid.aop/1024404286

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