## The Annals of Probability

- Ann. Probab.
- Volume 3, Number 6 (1975), 909-1068

### The Class of Subexponential Distributions

#### Abstract

The class $\mathscr{J}$ of subexponential distributions is characterized by $F(0) = 0, 1 - F^{(2)} (x) \sim 2\{1 - F(x)\}$ as $x \rightarrow \infty$. New properties of the class $\mathscr{J}$ are derived as well as for the more general case where $1 - F^{(2)} (x) \sim \beta\{1 - F(x)\}$. An application to transient renewal theory illustrates these results as does an adaptation of a result of Greenwood on randomly stopped sums of subexponentially distributed random variables.

#### Article information

**Source**

Ann. Probab. Volume 3, Number 6 (1975), 1000-1011.

**Dates**

First available: 19 April 2007

**Permanent link to this document**

http://projecteuclid.org/euclid.aop/1176996225

**JSTOR**

links.jstor.org

**Digital Object Identifier**

doi:10.1214/aop/1176996225

**Mathematical Reviews number (MathSciNet)**

MR391222

**Zentralblatt MATH identifier**

0374.60022

**Subjects**

Primary: 60E05: Distributions: general theory

Secondary: 60G40: Stopping times; optimal stopping problems; gambling theory [See also 62L15, 91A60]

**Keywords**

Subexponential distributions regular variation renewal theory branching process random sum

#### Citation

Teugels, Jozef L. The Class of Subexponential Distributions. The Annals of Probability 3 (1975), no. 6, 1000--1011. doi:10.1214/aop/1176996225. http://projecteuclid.org/euclid.aop/1176996225.