The harmonic mean formula for random processes

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.