On the Hitting Probability of Max-Stable Processes

TL;DR

This paper investigates the probability that max-stable processes in continuous function spaces hit specific points, establishing conditions for positive hitting probabilities and the likelihood of paths hitting points multiple times.

Contribution

It provides new insights into the hitting probabilities of max-stable processes and conditions under which paths can hit points multiple times.

Findings

Hitting probability is positive unless components are fully dependent.

Paths can hit points multiple times under certain conditions.

Hitting probabilities depend on the dependence structure of the process.

Abstract

The probability that a max-stable process {\eta} in C[0, 1] with identical marginal distribution function F hits x \in R with 0 < F (x) < 1 is the hitting probability of x. We show that the hitting probability is always positive, unless the components of {\eta} are completely dependent. Moreover, we consider the event that the paths of standard MSP hit some x \in R twice and we give a sufficient condition for a positive probability of this event.