Semi-Min-Stable Processes

Abstract

We define a semi-min-stable (SMS) process $Y(t)$ in $\lbrack 0,\infty)$ to be one which is stable under the simultaneous operations of taking the minima of $n$ independent copies of $Y(t)$ (pointwise over time $t$) and rescaling space and time. We show that the only possible rescaling of time is by a fixed power of $n$ and that SMS processes are essentially the only possible weak limits for large $m$ of a process obtained by taking the minimum, pointwise over $t$, of $m$ independent copies of a given process and then rescaling space and time. We describe the representation of a SMS process as the minimum of a Poisson process on a function space. We obtain a partial characterization of sample continuous SMS processes, similar to that of de Haan in the case of max-stable processes.