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Arcsine Laws and Interval Partitions Derived from a Stable Subordinator
Author(s) -
Pitman Jim,
Yor Marc
Publication year - 1992
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/plms/s3-65.2.326
Subject(s) - subordinator , mathematics , interval (graph theory) , pure mathematics , mathematical analysis , combinatorics , lévy process
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 B t ≠ 0 almost surely, and for t an inverse local time, when B t = 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.