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.