The harmonic mean formula for random processes

TL;DR

This paper extends the harmonic mean formula to general stochastically continuous random processes, exploring their supremum behavior and applications to distribution continuity and Pickands constants.

Contribution

It generalizes the harmonic mean formula for a broader class of random processes and discusses implications for supremum distribution and Pickands constants.

Findings

Extended harmonic mean formula for stochastically continuous processes

Established links between sojourn time and supremum of processes

Provided applications to distribution continuity and Pickands constants

Abstract

Motivated by the harmonic mean formula in [1], we investigate the relation between the sojourn time and supremum of a random process $X(t),t\in \mathbb{R}^d$ and extend the harmonic mean formula for general stochastically continuous $X$. We discuss two applications concerning the continuity of distribution of supremum of $X$ and representations of classical Pickands constants.