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April, 1991 Sample and Ergodic Properties of Some Min-Stable Processes
Keith Steven Weintraub
Ann. Probab. 19(2): 706-723 (April, 1991). DOI: 10.1214/aop/1176990447


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.


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Keith Steven Weintraub. "Sample and Ergodic Properties of Some Min-Stable Processes." Ann. Probab. 19 (2) 706 - 723, April, 1991.


Published: April, 1991
First available in Project Euclid: 19 April 2007

zbMATH: 0788.60050
MathSciNet: MR1106282
Digital Object Identifier: 10.1214/aop/1176990447

Primary: 60G17
Secondary: 60F99 , 60G10

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

Rights: Copyright © 1991 Institute of Mathematical Statistics

Vol.19 • No. 2 • April, 1991
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