The Annals of Probability

Nested classes of $C$-decomposable laws

John Bunge

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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

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

First available in Project Euclid: 18 June 2002

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

Class $L$ distribution decomposability semigroup infinite Bernoulli convolution infinitely divisible measure normed sum self-decomposable measure


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

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