Arcsine Laws and Interval Partitions Derived from a Stable Subordinator

Abstract

Lévy discovered that the fraction of time a standard one-dimensional Brownian motion B spends positive before time t has arcsine distribution, both for t a fixed time when Bt ≠ 0 almost surely, and for t an inverse local time, when Bt = 0 almost surely. This identity in distribution is extended from the fraction of time spent positive to a large collection of functionals derived from the lengths and signs of excursions of B away from 0. Similar identities in distribution are associated with any process whose zero set is the range of a stable subordinator, for instance a Bessel process of dimension d for 0 < d < 2.