## The Annals of Probability

- Ann. Probab.
- Volume 5, Number 1 (1977), 42-51.

### Random Stopping Preserves Regular Variation of Process Distributions

Priscilla Greenwood and Itrel Monroe

#### Abstract

Let $S_n$ be a stochastic process with either discrete or continuous time parameter and stationary independent increments. Let $N$ be a stopping time for the process such that $EN < \infty$. If the upper tail of the process distribution, $F$, is regularly varying, certain conditions on the lower tail of $F$ and on the tail of the distribution of $N$ imply that $\lim_{y\rightarrow\infty}P(S_N > y)/(1 - F(y)) = EN$. A similar asymptotic relation is obtained for $\sup_n S_{n \wedge N}$, if $n$ is discrete. These asymptotic results are related to the Wald moment identities and to moment inequalities of Burkholder. Applications are given for exit times at fixed and square-root boundaries.

#### Article information

**Source**

Ann. Probab., Volume 5, Number 1 (1977), 42-51.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176995889

**Digital Object Identifier**

doi:10.1214/aop/1176995889

**Mathematical Reviews number (MathSciNet)**

MR426139

**Zentralblatt MATH identifier**

0379.60043

**JSTOR**

links.jstor.org

**Subjects**

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

Secondary: 60J15 60J30

**Keywords**

Stopping times random walk independent increments regular variation asymptotics boundary crossing maximum process

#### Citation

Greenwood, Priscilla; Monroe, Itrel. Random Stopping Preserves Regular Variation of Process Distributions. Ann. Probab. 5 (1977), no. 1, 42--51. doi:10.1214/aop/1176995889. https://projecteuclid.org/euclid.aop/1176995889