## The Annals of Probability

- Ann. Probab.
- Volume 19, Number 2 (1991), 706-723.

### Sample and Ergodic Properties of Some Min-Stable Processes

#### Abstract

A random vector is $\min$-stable (or jointly negative exponential) if any weighted minimum of its components has a negative exponential distribution. The vectors can be subordinated to a two-dimensional homogeneous Poisson point process through positive $\mathscr{L}_1$ functions called spectral functions. A critical feature of this representation is the point of the Poisson process, called the location, that defines a $\min$-stable random variable. A measure of association between $\min$-stable random variables is used to define mixing conditions for $\min$-stable processes. The association between two $\min$-stable random variables $X_1$ and $X_2$ is defined as the probability that they share the same location and is denoted by $q(X_1, X_2).$ Mixing criteria for a $\min$-stable process $X(t)$ are defined by how fast the association between $X(t)$ and $X(t + s)$ goes to zero as $s \rightarrow \infty$. For some stationary processes (including the moving-minimum process), conditions on the spectral functions are derived in order that the processes satisfy mixing conditions.

#### Article information

**Source**

Ann. Probab., Volume 19, Number 2 (1991), 706-723.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176990447

**Digital Object Identifier**

doi:10.1214/aop/1176990447

**Mathematical Reviews number (MathSciNet)**

MR1106282

**Zentralblatt MATH identifier**

0788.60050

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60G17: Sample path properties

Secondary: 60G10: Stationary processes 60F99: None of the above, but in this section

**Keywords**

Min-stable random vectors min-stable processes two-dimensional Poisson process association

#### Citation

Weintraub, Keith Steven. Sample and Ergodic Properties of Some Min-Stable Processes. Ann. Probab. 19 (1991), no. 2, 706--723. doi:10.1214/aop/1176990447. https://projecteuclid.org/euclid.aop/1176990447