## The Annals of Probability

- Ann. Probab.
- Volume 6, Number 2 (1978), 279-293.

### Characterization of Subclasses of Class $L$ Probability Distributions

#### Abstract

The subclasses of class $L$ probability distributions recently studied by K. Urbanik are characterized by requiring that certain functions be convex and have derivatives of some fixed order. The extreme points of certain compact convex sets of probability measures are determined, and this information is then used to obtain a representation of the characteristic functions of the probability distributions in those classes, in the same manner as Urbanik has proceeded for the class $L$.

#### Article information

**Source**

Ann. Probab., Volume 6, Number 2 (1978), 279-293.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176995573

**Digital Object Identifier**

doi:10.1214/aop/1176995573

**Mathematical Reviews number (MathSciNet)**

MR471022

**Zentralblatt MATH identifier**

0409.60017

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60B15: Probability measures on groups or semigroups, Fourier transforms, factorization

Secondary: 60E05: Distributions: general theory 60F05: Central limit and other weak theorems 60G50: Sums of independent random variables; random walks 28A50: Integration and disintegration of measures

**Keywords**

Characteristic function Levy-Khincthine representation extreme points infinitely divisible distribution convex function completely monotonic function

#### Citation

Kumar, A.; Schreiber, B. M. Characterization of Subclasses of Class $L$ Probability Distributions. Ann. Probab. 6 (1978), no. 2, 279--293. doi:10.1214/aop/1176995573. https://projecteuclid.org/euclid.aop/1176995573