Occupation times and areas derived from random sampling

TL;DR

This paper introduces a novel combinatorial approach to analyze occupation times of spherical (fractional) Brownian motion, revealing uniform distribution properties and providing new proofs and characterizations of related stochastic process distributions.

Contribution

It presents a new combinatorial method for occupation times, offers an elementary proof of Lévy's second arcsine law, and characterizes occupation time distributions for all Lévy processes.

Findings

Occupation area of spherical (fractional) Brownian motion is uniformly distributed.

New combinatorial view effectively relates occupation times to persistence probabilities.

Elementary proof of Lévy's second arcsine law is provided.

Abstract

We consider the occupation area of spherical (fractional) Brownian motion, i.e. the area where the process is positive, and show that it is uniformly distributed. For the proof, we introduce a new simple combinatorial view on occupation times of stochastic processes that turns out to be surprisingly effective. A sampling method is used to relate the moments of occupation times to persistence probabilities of random walks that again relate to combinatorial factors in the moments of beta distributions. Our approach also yields a new and completely elementary proof of L\'evy's second arcsine law for Brownian motion. Further, combined with Spitzer's formula and the use of Bell polynomials, we give a characterisation of the distribution of the occupation times for all L\'evy processes.